Lesson 68 — Kinematics: Position, Velocity, and Acceleration

$s(t) = 2t^2 - 6t$. Calculate $v(t)$ and $a(t)$.

$s(t) = t^3 - 6t^2 + 9t$. When is $v = 0$? At each instant, is the object accelerating or braking?

$s(t) = 100 - 5t^2$ (free fall, $g = 10$ m/s²). When does it hit the ground? Velocity at that instant.

$s(t) = 5t + \frac{3}{2}t^2 + \frac{1}{2}t^3$. Velocity and acceleration at $t = 2$.

$s(t) = e^{-t}\sin t$. Calculate $v(t)$ and $a(t)$. What does the decaying amplitude reveal?

$s(t) = 10\sin(2t)$. Identify $A$, $\omega$, and the period $T$. Write $v(t)$.

$s(t) = t^4 - 4t^3 + 6t^2$. Maximum velocity on $[0, 3]$.

$s(t) = \ln(1 + t^2)$. Calculate $v(t)$ and evaluate at $t = 1$.

$s(t) = t^2 - 4t$. Distance traveled between $t = 0$ and $t = 4$ (note: $v$ changes sign).

$s(t) = A\cos(\omega t)$. Calculate the jerk $j(t) = s'''(t)$.

$s(t) = 2t^3 - 6t + 1$. When is velocity zero? Is there a change in direction?

$s(t) = \sin(t^2)$. Calculate $v(t)$ (chain rule) and evaluate at $t = \sqrt{\pi}$.

A ball is thrown upward from the ground with $v_0 = 20$ m/s. Maximum height ($g = 10$ m/s²).

A car at $v_0 = 30$ m/s brakes uniformly with $a = -5$ m/s². Stopping distance (Torricelli's Theorem).

An airplane starts from rest and takes off at $v_f = 80$ m/s after a runway of $1000$ m. Average acceleration and takeoff time.

A stone falls from $h = 80$ m. Time to fall and speed at impact ($g = 10$ m/s²).

A car accelerates from 0-100 km/h in $10.5$ s. Average acceleration and distance traveled during acceleration.

Projectile motion: $v_0 = 50$ m/s at $30°$ to the horizontal. Horizontal range ($g = 10$ m/s²).

Rocket: $a(t) = 30 - 0.5t$ m/s² until $t = 60$ s (engine cutoff). Velocity and position at engine cutoff.

A train brakes uniformly, traveling $200$ m in $20$ s and stopping. What was its initial $v_0$?

A ball is thrown from the top of a $50$ m tower with $v_0 = 20$ m/s upwards. Time until it hits the ground.

An object of $m = 1$ kg falls with drag $b = 0.2$ kg/s. Terminal velocity ($g = 10$ m/s²).

Mass-spring system: $m = 1$ kg, $k = 100$ N/m. Angular frequency $\omega$, period $T$, and frequency $f$.

$x(t) = 0.1\cos(2\pi t)$. Amplitude, period, $v(t)$, and maximum velocity.

Pendulum of length $L = 1$ m. Angular frequency $\omega = \sqrt{g/L}$ and period ($g = 10$ m/s²).

Verify that $x(t) = A\cos(\omega t + \phi)$ satisfies the ODE $\ddot{x} + \omega^2 x = 0$.

SHM: $E = \frac{1}{2}m\dot{x}^2 + \frac{1}{2}kx^2$. Show that $E$ is constant by differentiating with respect to time.

$x(t) = e^{-t}\cos(5t)$ (damped oscillator). Apparent frequency and behavior of the amplitude.

Phase relationship between $x(t) = A\cos(\omega t + \phi)$ and $v(t)$. Confirm $90°$.

Show that $a(t)$ and $x(t)$ are $180°$ out of phase in SHM — i.e., $a = -\omega^2 x$.

A ball is thrown upward. At its highest point, the acceleration is:

Explain why average velocity ($\Delta s/\Delta t$) $\neq$ average of velocities in general. Give a numerical example.

Explain the difference between velocity (1D vector quantity with sign) and speed (scalar). Why is $v < 0$ possible?

Circular motion: $\vec{r} = R(\cos\omega t, \sin\omega t)$. Show that $\vec{a} = -\omega^2\vec{r}$ and $|\vec{a}| = R\omega^2$.

Projectile motion launched with $v_0$ and angle $\theta$. Derive the range formula $R = v_0^2\sin(2\theta)/g$ and the optimal angle.

Car travels at 60 km/h for 1 hour, then at 120 km/h for 1 hour. Average velocity by time? Average velocity over equal distances traveled?

Quadratic drag: $m\dot{v} = -mg - bv^2$. Terminal velocity and analytical solution for $v(t)$ (via separation of variables).

Helix: $\vec{r}(t) = (R\cos\omega t, R\sin\omega t, vt)$. Calculate $\vec{v}$, $|\vec{v}|$, and $\vec{a}$.

Derive Torricelli's equation $v_f^2 = v_0^2 + 2a\,\Delta s$ from the UAM equations by eliminating time $t$.

Show that in SHM, the time-averaged kinetic and potential energies are each $E/2$ — using $\langle\sin^2\rangle = \langle\cos^2\rangle = 1/2$.