What is Velocity and acceleration? — Physics

Definition 11.2 University Physics — Year 1 · Chapter 11 — Kinematics of a Point

The velocity and acceleration of $M$ in the frame are

\vect v = \frac{\dd\vect{OM}}{\dd t}, \qquad \vect a = \frac{\dd\vect v}{\dd t} = \frac{\dd^2\vect{OM}}{\dd t^2},

derivatives of vector functions: the derivative of a vector is obtained by differentiating its components in a basis that is fixed in the frame. The velocity is tangent to the trajectory and points along the motion; its norm $v$ is the speed. Units: $\mathrm{m}/\mathrm{s}$ , $\mathrm{m}/\mathrm{s}^{2}$ .

A projectile’s parabola: the velocity is tangent to the trajectory and turns, the acceleration g is constant. At the apex velocity and acceleration are perpendicular: the speed is momentarily stationary while the direction still changes. — A projectile’s parabola: the velocity is tangent to the trajectory and turns, the acceleration $\vect g$ is constant. At the apex velocity and acceleration are perpendicular: the speed is momentarily stationary while the direction still changes.

Examples

Example 11.4 (Projectile)

Launched from $O$ at speed $v_0$ and angle $\alpha$ above the horizontal, a ball’s coordinates (dynamics, next chapter) are $x = v_0\cos\alpha\,t$ , $z = v_0\sin\alpha\,t - \tfrac12 gt^2$ . Then $\vect v = v_0\cos\alpha\,\vect e_x + (v_0\sin\alpha - gt)\vect e_z$ , $\vect a = -g\vect e_z$ : the acceleration is constant while the velocity turns; at the apex ( $\dot z = 0$ , $t = v_0\sin\alpha/g$ ) the velocity is horizontal and the acceleration perpendicular to it. Eliminating $t$ : $z = x\tan\alpha - gx^2/(2v_0^2\cos^2\alpha)$ , a parabola.

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Examples

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