---
title: "Forces and Motion"
book: "High School Physics"
subject: physics
language: en
chapter: 16
exercises: 15
source: https://one-course.com/books/physics/2/en/chapter/16-forces-and-motion
---

# Chapter 16 — Forces and Motion

Throw a wrench spinning across the room: the ends wobble wildly, yet one point inside it traces a perfect parabola. [Chapter 6](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#ch-g10-inertia) settled what happens when [forces](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) [compensate](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-compensate) — nothing. This chapter asks what happens when they do not: the leftover [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) *changes the velocity* — speeding bodies up, slowing them down, turning them — and says by how much, in ratios if not yet in formulas.

## 16.1 The center of mass

**Definition 16.1 (Center of mass).**

The *center of mass* of a body or system is its balance point — the average position of its mass: the geometric center of a homogeneous symmetric body; for two point masses, the point of the joining segment at distances in the *inverse* ratio of the masses, $\frac{d_1}{d_2} = \frac{m_2}{m_1}$ — closer to the heavier.

**Example 16.2 (A lopsided dumbbell).**

Balls of $2.0\,\mathrm{kg}$ and $1.0\,\mathrm{kg}$ sit at the ends of a light bar $0.90\,\mathrm{m}$ long. The [center of mass](#def-g11-forces-and-motion-com) divides it in the inverse ratio $d_1/d_2 = 1.0/2.0$: $0.30\,\mathrm{m}$ from the heavy ball, $0.60\,\mathrm{m}$ from the light one — balance the bar there.

**Remark 16.3 (The thrown wrench).**

Film a tumbling wrench: every point follows a complicated looping curve — except the [center of mass](#def-g11-forces-and-motion-com), which traces the clean parabola a thrown ball would. The motion of the [center of mass](#def-g11-forces-and-motion-com) is simple; the rest is spin *around* it — which is why mechanics may treat a car, a planet or a gymnast as a single point.

![A thrown wrench at equal time intervals: the wrench spins, but its center of mass (dots) rides a plain parabola.](https://one-course.com/images/onecourse/chapters/physics-2/g11-forces-and-motion/fig-d744acf3fecb.svg)

*A thrown wrench at equal time intervals: the wrench spins, but its [center of mass](#def-g11-forces-and-motion-com) (dots) rides a plain parabola.*

## 16.2 A net force changes the velocity

**Definition 16.4 (Net force).**

The *net force* on a body is the vector sum of all the [forces](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) acting on it, $\vect F = \vect F_1 + \vect F_2 + \dots$ (tip to tail, as in the mathematics volume); the [forces](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) [compensate](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-compensate) exactly when $\vect F = \vect 0$.

**Theorem 16.5 (Effect of a net force — semi-quantitative).**

A [net force](#def-g11-forces-and-motion-netforce) $\vect F$ acting on a body of mass $m$ for a short duration $\Delta t$ changes the velocity of its [center of mass](#def-g11-forces-and-motion-com) by a vector $\Delta \vect v$ that has the *direction* of $\vect F$, and whose magnitude grows in proportion to $F$ and to $\Delta t$ and shrinks in proportion to the mass:

$$
\Delta v \propto \frac{F \, \Delta t}{m} .
$$

**Proof.** *Admitted at this level.* ∎

**Remark 16.6.**

Double the [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) or its duration: twice the change; double the mass: half. Next year sharpens this into an exact equation, and the Year 1 volume asks in which frames it holds; this year ratios are enough — they already run airbags, brakes and [orbits](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-satellite).

**Proposition 16.7 (The three canonical cases).**

Let a body move with velocity $\vect v$ under a [net force](#def-g11-forces-and-motion-netforce) $\vect F$.

1. $\vect F$ along $\vect v$ : it *speeds up* in a straight line;
2. $\vect F$ opposite $\vect v$ : it *slows down* in a straight line;
3. $\vect F$ perpendicular to $\vect v$ : it *turns* without changing speed.

**Proof.** Add $\Delta\vect v$, directed along $\vect F$ ([Theorem 16.5](#thm-g11-forces-and-motion-secondlaw)), tip to tail onto $\vect v$: along $\vect v$ the arrows pile up (longer); opposite, they partly cancel (shorter); perpendicular, each small $\Delta \vect v$ tilts the arrow without lengthening it — only the direction turns. ∎

![The three canonical cases: the net force drags the tip of the velocity vector its own way — longer, shorter, or around.](https://one-course.com/images/onecourse/chapters/physics-2/g11-forces-and-motion/fig-c16373ab40b3.svg)

![The three canonical cases: the net force drags the tip of the velocity vector its own way — longer, shorter, or around.](https://one-course.com/images/onecourse/chapters/physics-2/g11-forces-and-motion/fig-90e75667a6a1.svg)

![The three canonical cases: the net force drags the tip of the velocity vector its own way — longer, shorter, or around.](https://one-course.com/images/onecourse/chapters/physics-2/g11-forces-and-motion/fig-cc9b8970783f.svg)

*The three canonical cases: the [net force](#def-g11-forces-and-motion-netforce) drags the tip of the velocity vector its own way — longer, shorter, or around.*

**Example 16.8 (A sideways kick).**

A puck glides east at $6.0\,\mathrm{m}/\mathrm{s}$; a brief sideways blow adds $\Delta\vect v = 8.0\,\mathrm{m}/\mathrm{s}$ due north. The new velocity is the vector sum: magnitude $\sqrt{6.0^2 + 8.0^2} = 10.0\,\mathrm{m}/\mathrm{s}$, direction $\tan\theta = 8.0/6.0$, i.e. $\theta \approx 53^\circ$ north of east — the old velocity is not erased, the change is added to it.

## 16.3 Comparing changes: force, time, mass

**Method 16.9 (Ratio reasoning).**

To compare two situations, never solve an equation — form ratios: the velocity-change ratio is the $F$ ratio, times the $\Delta t$ ratio, divided by the $m$ ratio. Then read off which factor changed, by how much.

**Example 16.10 (The loaded cart).**

The same push, applied for the same second, sends an empty $20\,\mathrm{kg}$ shopping cart off at $1.2\,\mathrm{m}/\mathrm{s}$. Loaded to $60\,\mathrm{kg}$ — triple mass, same $F\Delta t$ — it leaves at a third of that, $0.40\,\mathrm{m}/\mathrm{s}$. Mass is the stubbornness of matter.

**Remark 16.11 (Crashes, airbags, bent knees).**

In a crash $\Delta v$ and $m$ are not negotiable — the occupant’s speed must reach zero — so $F\,\Delta t$ is fixed. The only freedom left is *time*: stretch $\Delta t$ and the [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) shrinks in the same ratio. A head stopping on an airbag in $0.08\,\mathrm{s}$ instead of on the wheel in $0.01\,\mathrm{s}$ feels an average [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) eight times smaller; crumple zones, helmet foam and bent knees play the same trick.

## 16.4 Friction: the force that fights sliding

**Definition 16.12 (Static and kinetic friction).**

[Friction](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-inventory) between touching surfaces opposes their *relative sliding*. *Static friction* acts while they do not slide, adjusting itself up to a maximum to cancel whatever tries to start the slide; *kinetic friction*, once sliding has begun, is roughly constant, directed against the sliding, and usually *weaker* than the static maximum.

**Example 16.13 (Pushing the wardrobe).**

Push a wardrobe gently: it does not move — static [friction](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-inventory) cancels your $150\,\mathrm{N}$. Push harder: at some $400\,\mathrm{N}$ it lurches into motion, then slides with less effort — the weaker [kinetic friction](#def-g11-forces-and-motion-friction) has taken over.

**Remark 16.14 (Why ABS brakes work).**

A rolling wheel’s contact point does not slide on the road: the tyre grips with *static* [friction](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-inventory). A locked wheel skids and gets only the weaker *kinetic* [friction](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-inventory), which points against the skid whatever the wheels do — steering dies. ABS releases the brake the instant a wheel locks: more grip, shorter stop, a driver who can steer.

![Typical braking distances from 90\, km/ h: the tyre–road friction is the only force slowing the car, and water or snow cuts it badly.](https://one-course.com/images/onecourse/chapters/physics-2/g11-forces-and-motion/fig-59f6b494c2b3.svg)

*Typical braking distances from $90\,\mathrm{km}/\mathrm{h}$: the tyre–road [friction](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-inventory) is the only [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) slowing the car, and water or snow cuts it badly.*

## 16.5 Turning: circular motion needs an inward force

**Definition 16.15 (Uniform circular motion).**

A body is in *uniform circular motion* when its [trajectory](https://one-course.com/books/physics/2/en/chapter/5-relative-motion#def-g10-relative-motion-trajectory) is a circle traveled at constant speed. The velocity, tangent to the circle, changes direction at every instant — so it is never constant.

**Proposition 16.16 (The inward force).**

A body in [uniform circular motion](#def-g11-forces-and-motion-ucm) is subject to a [net force](#def-g11-forces-and-motion-netforce) that is never zero: at every instant it points from the body toward the center of the circle.

**Proof.** The speed is constant, so the [net force](#def-g11-forces-and-motion-netforce) has no part along $\vect v$ (cases 1 and 2 of [Proposition 16.7](#prop-g11-forces-and-motion-cases)): it is perpendicular to the tangent, hence along the radius — and inward, since every $\Delta \vect v$ bends the path toward the center. ∎

![A ball whirled on a string: the velocity is tangent, the pull points to the center. Cut the string: the ball leaves along the tangent.](https://one-course.com/images/onecourse/chapters/physics-2/g11-forces-and-motion/fig-0074a12a6ec1.svg)

*A ball whirled on a string: the velocity is tangent, the pull points to the center. Cut the string: the ball leaves along the tangent.*

**Example 16.17 (The string, the bend, the Moon).**

Every steady turn hides an inward [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force). The whirled ball: the string’s [tension](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-inventory). The car on a bend: the sideways static [friction](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-inventory) of road on tyres — gone on ice, and the car goes straight. The Moon: the Earth’s gravitational pull ([Chapter 4](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#ch-g10-universal-gravitation)), turning it around in $27.3$ days — not held *up* but pulled sideways into a perpetual turn, falling around the Earth; next year makes this the mechanics of [satellites](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-satellite).

## 16.6 Exercises

**Exercise 16.1 ★.**

Zero or nonzero [net force](#def-g11-forces-and-motion-netforce)? Justify from the velocity: (a) a car cruising straight at a steady $90\,\mathrm{km}/\mathrm{h}$; (b) braking in a straight line; (c) rounding a bend at a steady $50\,\mathrm{km}/\mathrm{h}$; (d) a skydiver at terminal speed.

**Solution of Exercise 16.1.**

(a) Constant velocity: [net force](#def-g11-forces-and-motion-netforce) zero. (b) Speed changes: nonzero, opposite the motion. (c) Direction changes (velocity is a vector): nonzero, toward the inside of the bend. (d) Constant velocity: zero — [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight) and drag [compensate](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-compensate).

**Exercise 16.2 ★.**

Balls of $3.0\,\mathrm{kg}$ and $1.0\,\mathrm{kg}$ sit at the ends of a light bar $0.80\,\mathrm{m}$ long. Locate the [center of mass](#def-g11-forces-and-motion-com); where for equal masses?

**Solution of Exercise 16.2.**

$d_1/d_2 = 1.0/3.0$ with $d_1 + d_2 = 0.80\,\mathrm{m}$: $d_1 = 0.20\,\mathrm{m}$ from the $3.0\,\mathrm{kg}$ ball. Equal masses: the middle, $0.40\,\mathrm{m}$ from each.

**Exercise 16.3 ★.**

The same brief push is given to an empty $15\,\mathrm{kg}$ luggage trolley and to the same trolley loaded to $45\,\mathrm{kg}$. Compare the velocity changes; what push would give the loaded trolley the empty one’s $\Delta v$?

**Solution of Exercise 16.3.**

Same $F\Delta t$, triple mass: $\Delta v$ three times smaller loaded. To match the empty trolley’s $\Delta v$, push three times harder (or three times longer).

**Exercise 16.4 ★.**

A tram moving at $10\,\mathrm{m}/\mathrm{s}$ brakes; one second later it moves at $8\,\mathrm{m}/\mathrm{s}$, same direction. Give $\Delta \vect v$ and the direction of the [net force](#def-g11-forces-and-motion-netforce). Which case of [Proposition 16.7](#prop-g11-forces-and-motion-cases) is this?

**Solution of Exercise 16.4.**

$\Delta \vect v$: $2\,\mathrm{m}/\mathrm{s}$, directed *backward* (from $10$ to $8\,\mathrm{m}/\mathrm{s}$). The [net force](#def-g11-forces-and-motion-netforce) points the same way: opposite $\vect v$ — case 2, slowing down.

**Exercise 16.5 ★.**

Static or [kinetic friction](#def-g11-forces-and-motion-friction) — and directed which way? (a) A crate resisting your (too gentle) push; (b) the same crate once it slides; (c) a book on a slightly tilted tray, not sliding.

**Solution of Exercise 16.5.**

(a) Static, opposite your push (no sliding yet). (b) Kinetic, opposite the sliding. (c) Static, pointing *up* the slope — against the slide that gravity is trying to start.

**Exercise 16.6 ★★.**

A [net force](#def-g11-forces-and-motion-netforce) $F$ applied for $\Delta t$ to a mass $m$ produces $\Delta v = 4.0\,\mathrm{m}/\mathrm{s}$. Predict $\Delta v$ for: (a) $2F, \Delta t, m$; (b) $F, \Delta t/2, m$; (c) $F, \Delta t, 2m$; (d) $3F, 2\Delta t, 6m$.

**Solution of Exercise 16.6.**

$\Delta v \propto F\Delta t/m$: (a) $8.0\,\mathrm{m}/\mathrm{s}$; (b) $2.0\,\mathrm{m}/\mathrm{s}$; (c) $2.0\,\mathrm{m}/\mathrm{s}$; (d) $4.0 \times \frac{3 \times 2}{6} =
4.0\,\mathrm{m}/\mathrm{s}$ — the three changes cancel.

**Exercise 16.7 ★★.**

A puck slides east at $8.0\,\mathrm{m}/\mathrm{s}$; a blow adds $\Delta \vect v = 6.0\,\mathrm{m}/\mathrm{s}$ due north. Compute the new speed and direction. Why is the speed not $8.0 + 6.0 = 14\,\mathrm{m}/\mathrm{s}$?

**Solution of Exercise 16.7.**

$v' = \sqrt{8.0^2 + 6.0^2} = 10.0\,\mathrm{m}/\mathrm{s}$, at $\tan\theta =
6.0/8.0$, i.e. $\theta \approx 37^\circ$ north of east. Velocities add as *vectors*: perpendicular contributions combine by Pythagoras, not by addition of magnitudes.

**Exercise 16.8 ★★.**

A goalkeeper stops the same shot two ways: arms rigid (the ball stops in $0.02\,\mathrm{s}$) or “giving” with the ball ($0.08\,\mathrm{s}$). Compare the average [forces](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force), and state which quantities were fixed before she chose.

**Solution of Exercise 16.8.**

Fixed beforehand: the ball’s mass and its $\Delta v$ (incoming speed to zero), hence the product $F\Delta t$. Stretching $\Delta t$ from $0.02\,\mathrm{s}$ to $0.08\,\mathrm{s}$ divides the average [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) by $4$.

**Exercise 16.9 ★★.**

Which external [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) pushes a sprinter forward at the start? Why do spiked shoes help — and why can nobody accelerate, or even walk, on perfectly frictionless ice?

**Solution of Exercise 16.9.**

The static [friction](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-inventory) of the track on the shoe: the foot pushes backward on the ground, and the mutual [contact force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-contact) pushes the sprinter forward. Spikes raise the maximum static [friction](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-inventory) (no slip at full effort). On frictionless ice no external horizontal [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) exists, so the velocity of the [center of mass](#def-g11-forces-and-motion-com) cannot change — no start, no walk.

**Exercise 16.10 ★★.**

In an emergency stop, why does a car with ABS (wheels kept rolling) usually stop shorter than one with locked, skidding wheels — and why can the skidding car no longer be steered? Quote [Definition 16.12](#def-g11-forces-and-motion-friction).

**Solution of Exercise 16.10.**

A rolling wheel’s contact point does not slide: static [friction](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-inventory), stronger than the [kinetic friction](#def-g11-forces-and-motion-friction) of a skid ([Definition 16.12](#def-g11-forces-and-motion-friction)) — shorter stop. A skidding tyre’s [kinetic friction](#def-g11-forces-and-motion-friction) points opposite the sliding, whatever the wheels’ angle: turning the wheel changes nothing, steering is lost.

**Exercise 16.11 ★★.**

A hammer thrower whirls the ball on its wire. What [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) turns the ball? Sketch its path after release. Why does the Moon need no wire, and what would its path become without gravity?

**Solution of Exercise 16.11.**

The [tension](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-inventory) of the wire, pointing inward to the athlete. After release the ball flies off in a straight line along the tangent, at the speed it had (inertia). The Moon’s “wire” is the Earth’s gravitational pull; switch gravity off and it leaves along its tangent, straight and uniform, forever.

**Exercise 16.12 ★★★.**

A broom is a $1.0\,\mathrm{kg}$ uniform stick (own [center of mass](#def-g11-forces-and-motion-com) $0.60\,\mathrm{m}$ from the top) with a $2.0\,\mathrm{kg}$ head whose [center of mass](#def-g11-forces-and-motion-com) is $1.20\,\mathrm{m}$ from the top. Treating each part as a point mass, locate the broom’s [center of mass](#def-g11-forces-and-motion-com) and check the inverse-ratio rule of [Definition 16.1](#def-g11-forces-and-motion-com).

**Solution of Exercise 16.12.**

From the top: $x = \dfrac{1.0 \times 0.60 + 2.0 \times 1.20}{3.0}
= 1.00\,\mathrm{m}$. Distances to the two part-centers: $0.40\,\mathrm{m}$ (stick) and $0.20\,\mathrm{m}$ (head), ratio $2 : 1$ — the inverse of the mass ratio $1.0 : 2.0$, as [Definition 16.1](#def-g11-forces-and-motion-com) requires.

**Exercise 16.13 ★★★.**

A car of mass $m$ and a loaded van of mass $2m$ brake from the same speed with the same braking [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force).

1. Compare the times needed to stop.
2. Each loses speed steadily, hence travels at the same *average* speed: compare the stopping distances.
3. The car now brakes from *double* the speed, same [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) : compare time and distance with its first stop. Moral for speed limits?

**Solution of Exercise 16.13.**

*1.* Same $F$ and $\Delta v$, double $m$: the van needs twice the time. *2.* Same [average speed](https://one-course.com/books/physics/2/en/chapter/5-relative-motion#def-g10-relative-motion-speed) for twice the time: twice the distance. *3.* Double $\Delta v$ at the same $F$ and $m$: twice the time; but the [average speed](https://one-course.com/books/physics/2/en/chapter/5-relative-motion#def-g10-relative-motion-speed) also doubles, so the distance is multiplied by $4$. Doubling your speed quadruples your braking distance.

**Exercise 16.14 ★★★.**

The Moon [orbits](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-satellite) at $R \approx 3.84 \times 10^{8}\,\mathrm{m}$ in $T = 27.3$ days.

1. Compute its orbital speed (circumference over [period](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-period) ).
2. Which way does its velocity change point, and which [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) provides it?
3. Improve, using [Proposition 16.16](#prop-g11-forces-and-motion-inward) : “the Moon does not fall because it moves fast enough”.

**Solution of Exercise 16.14.**

*1.* $T = 27.3 \times 86\,400\,\mathrm{s} \approx 2.36 \times 10^{6}\,\mathrm{s}$, so $v = \dfrac{2\pi \times 3.84 \times 10^{8}}{2.36 \times 10^{6}} \approx
1.0 \times 10^{3}\,\mathrm{m}/\mathrm{s} \approx 1\,\mathrm{km}/\mathrm{s}$. *2.* Toward the Earth (inward); the Earth’s gravitational pull. *3.* The Moon *is* falling: its velocity change points earthward at every instant. Its tangential speed merely ensures that it keeps missing — it falls *around* the Earth instead of into it.

**Exercise 16.15 ★★★.**

An egg of mass $50\,\mathrm{g}$ hits the floor at $4.0\,\mathrm{m}/\mathrm{s}$: on tile it stops in about $1\,\mathrm{ms}$, on thick foam in about $50\,\mathrm{ms}$. Compare the average [forces](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force). Explain, with the same reasoning, why you bend your knees when landing a jump, and what a gymnast’s crash mat is made of.

**Solution of Exercise 16.15.**

Same $m$ and $\Delta v$, so $F\Delta t$ is fixed: $\Delta t$ fifty times longer on foam means an average [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) fifty times smaller. Bending the knees stretches the landing’s $\Delta t$ from a jolt to a long flex; a crash mat is, mechanically speaking, made of extra milliseconds.

## 16.7 Problem: The crash test

**Problem 16.1.**

Weekend problem — designing the gentle collision: since a crash fixes the velocity change, every safety device in a car is a machine for stretching time

A crash-test laboratory drives a car at $50\,\mathrm{km}/\mathrm{h}$ into a rigid wall, with a $70\,\mathrm{kg}$ adult dummy and a $20\,\mathrm{kg}$ child dummy aboard; cameras time every stop. Your job: find where the violence of a crash lives, and design it away.

**Part I — The measure of a crash.**

1. Convert $50\,\mathrm{km}/\mathrm{h}$ to $\mathrm{m}/\mathrm{s}$ .
2. In any frontal stop the occupants’ velocity change has the same magnitude. Which one? Why can no device alter it?
3. Quote [Theorem 16.5](#thm-g11-forces-and-motion-secondlaw) : with $\Delta v$ and $m$ fixed, $F\,\Delta t$ is fixed. The designer’s only dial?
4. The wall stops the car in $0.050\,\mathrm{s}$ ; a crumple zone stretches this to $0.150\,\mathrm{s}$ . Compare the average [forces](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) .
5. In which direction do $\Delta \vect v$ and the [net force](#def-g11-forces-and-motion-netforce) on the dummies point during the stop?

**Part II — The crumple zone.**

6. Why, in one sentence, do engineers design the front of the car to be destroyed?
7. In a vintage rigid car (stop in $0.050\,\mathrm{s}$ ) the dummy is strapped rigidly to the seat. Compare its [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) with the modern $0.150\,\mathrm{s}$ stop.
8. The stopping car’s speed falls steadily (average: half the initial). How far does it travel in $0.150\,\mathrm{s}$ ? Compare with a real crumple zone.
9. Why not a $10\,\mathrm{m}$ crumple zone? And which part of the car must *not* crumple?
10. Summarize Part II as a slogan: “you cannot choose $\Delta v$ , so you must choose …”.

**Part III — The belt, the airbag and the child.**

11. An unbelted dummy keeps its $14\,\mathrm{m}/\mathrm{s}$ ( [Chapter 6](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#ch-g10-inertia) ) until the windshield stops it in $0.005\,\mathrm{s}$ ; a belted one stops with the car in $0.150\,\mathrm{s}$ . Compare the average [forces](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) .
12. Why is a belt that stretches slightly better than an unstretchable steel harness?
13. The head would stop on the wheel in $0.010\,\mathrm{s}$ ; the airbag stretches this to $0.080\,\mathrm{s}$ . Compare the [forces](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) .
14. Adult and child undergo the same $\Delta v$ in the same $\Delta t$ . Compare the [net forces](#def-g11-forces-and-motion-netforce) the two belts must supply.
15. A child on a lap: the arms must change a $10\,\mathrm{kg}$ baby’s velocity by $14\,\mathrm{m}/\mathrm{s}$ in $0.150\,\mathrm{s}$ , while gravity changes a velocity by $9.8\,\mathrm{m}/\mathrm{s}$ each second. How many times faster must the arms act — holding what mass would feel that heavy? Conclude.
16. Children feel smaller [forces](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) — so why child seats? Think of *where* the belt applies its [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) , and of $\Delta t$ .

**Part IV — The safety cage, and the verdict.**

17. The passenger cell is rigid while both ends crumple. Why must the cell not deform, though rigidity means violent stops?
18. Two identical cars collide head-on, each at $50\,\mathrm{km}/\mathrm{h}$ . Where does each stop? Compare each crash with the wall test.
19. A small car meets a loaded truck head-on. The [forces](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) they exert on each other are equal and opposite, mutual as every interaction is ( [Chapter 13](https://one-course.com/books/physics/2/en/chapter/13-the-fundamental-interactions#ch-g11-fundamental-interactions) ). Whose velocity changes more?
20. List the safety devices of this problem and the single quantity every one of them stretches.
21. Finale: state the design law of the gentle collision in one sentence using only *mass* , *velocity change* , *[force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force)* and *time* .

**Solution of Problem 16.1.**

**1.** $50/3.6 \approx 13.9\,\mathrm{m}/\mathrm{s}$ (call it $14\,\mathrm{m}/\mathrm{s}$). **2.** $\Delta v \approx 14\,\mathrm{m}/\mathrm{s}$: the occupant moves at $14\,\mathrm{m}/\mathrm{s}$ before and at zero after — both ends are fixed by the crash, so no device can touch $\Delta v$. **3.** $\Delta v \propto F\Delta t/m$ ([Theorem 16.5](#thm-g11-forces-and-motion-secondlaw)) with $\Delta v$ and $m$ fixed [forces](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) $F\Delta t$ to be fixed. The designer’s only dial is $\Delta t$. **4.** $0.150/0.050 = 3$: the crumple zone divides the average [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) by $3$. **5.** Backward, opposite the motion — the velocity shrinks from $14\,\mathrm{m}/\mathrm{s}$ to zero. **6.** The folding front is a machine for making the stop last longer: metal that crumples buys milliseconds. **7.** Same $\Delta v$ and $m$, $\Delta t$ three times shorter: the vintage dummy takes three times the [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force). **8.** [Average speed](https://one-course.com/books/physics/2/en/chapter/5-relative-motion#def-g10-relative-motion-speed) $\approx 6.9\,\mathrm{m}/\mathrm{s}$; distance $\approx
6.9 \times 0.150 \approx 1.0\,\mathrm{m}$ — just about the [metre](https://one-course.com/books/physics/2/en/chapter/1-orders-of-magnitude-measuring-the-universe#def-g10-orders-of-magnitude-unit) of deformable car a real crumple zone provides. **9.** A $10\,\mathrm{m}$ nose is undrivable and unparkable, and its own mass would worsen every crash. The passenger cell must not crumple: it preserves the survival space. **10.** “…so you must choose $\Delta t$.” **11.** $0.150/0.005 = 30$: the windshield stop is thirty times more violent than the belted one. **12.** A slightly stretching belt lengthens the occupant’s own $\Delta t$ a little more (and spreads the [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force)); a steel harness would impose the car body’s shortest stop on the softest passenger. **13.** $0.080/0.010 = 8$: the airbag divides the head’s [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) by $8$. **14.** Same $\Delta v$ and $\Delta t$: the [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) scales with the mass, $70/20 = 3.5$ times larger for the adult. **15.** Required change: $14/0.150 \approx 93\,\mathrm{m}/\mathrm{s}$ each second — about $9.5$ times what gravity produces ($9.8\,\mathrm{m}/\mathrm{s}$ per second). The arms must pull about $9.5$ times the baby’s [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight): like holding a $95\,\mathrm{kg}$ load. No arms can; the baby flies forward. **16.** The belt of an adult routes [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) to hips and chest; on a child it would load belly and neck. The child seat applies the (smaller) [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) to strong body parts and its padding stretches $\Delta t$ further. **17.** If the cell deforms, the occupants lose their survival space and are struck by the structure itself. Nothing should hit the rigid cell directly: the occupants’ stops are handled by belt and airbag (long $\Delta t$), while the cell anchors the crumpling ends. **18.** By symmetry both cars stop at the contact plane: each driver undergoes $\Delta v \approx 14\,\mathrm{m}/\mathrm{s}$ over a crumple-zone stop — essentially the wall test, not “double the crash”. **19.** The [forces](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) are equal and opposite, but $\Delta v \propto F\Delta t/m$: the small car, with the small $m$, suffers the large velocity change. **20.** Crumple zone, stretching seat belt, airbag, child-seat padding — every one of them stretches the same quantity: $\Delta t$. **21.** The crash fixes the mass and the velocity change, hence the product [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) $\times$ time: a gentle collision is a long one — stretch the time, and the [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) falls in the same ratio.