Kinematics: notes and practice questions

If the velocity, $v\ ms^{- 1}$, of a particle which is moving in a straight line after time, $t$ seconds, is $v(t) = 2\cos(t) - \sin(3t)$ for $0 \leq t \leq 4$.

$a$ Find the maximum speed.

$b$ At what time there would be a change in direction?

$c$ Find acceleration after 3 seconds

$d$ Find the total travelled distance.

The acceleration, $a \text{ ms}^{-2}$, of a particle moving in a straight line at time $t$ seconds, $t \ge 0$, is given by $a = -2v - 8$, where $v \text{ ms}^{-1}$ is the particle's velocity. At $t=0$, the particle is at the origin O and has an initial velocity $v_0 \text{ ms}^{-1}$, where $v_0 > 0$.

By solving an appropriate differential equation, show that the particle's velocity at time $t$ is given by $v(t) = (v_0 + 4)e^{-2t} - 4$.

The particle moves in the positive direction until it reaches its maximum displacement from O at time $T$. Show that $e^{2T} = \frac{v_0+4}{4}$.

Find an expression for the maximum displacement, $s_{\text{max}}$, in terms of $v_0$.

Let $v(T-k)$ represent the particle's velocity $k$ seconds before it reaches $s_{\text{max}}$, where $0 < k < T$. By using the result from part (b)(i), show that $v(T-k) = 4(e^{2k} - 1)$.

Similarly, let $v(T+k)$ represent the particle's velocity $k$ seconds after it reaches $s_{\text{max}}$. Deduce a similar expression for $v(T+k)$ in terms of $k$.

Hence, show that the speed of the particle $k$ seconds before it reaches $s_{\text{max}}$ is greater than or equal to its speed $k$ seconds after it reaches $s_{\text{max}}$.

A small reconnaissance drone is performing a vertical ascent and descent. Its vertical velocity, $v \text{ ms}^{-1}$, at time $t$ seconds, for $0 \le t \le 10$, is modelled by the function $v(t) = t \sin t - 2.5$.

The following diagram shows the graph of $v$.

(a) Find the smallest value of $t$ for which the drone is momentarily stationary.

(b) Find the total vertical distance travelled by the drone during the first 10 seconds.

(c) Find the vertical acceleration of the drone when $t=8$ seconds.

A drone takes off from a platform. Its height, $h$ metres, above the platform after $t$ seconds is given by $h(t) = 6t - t^2$, for $0 \le t \le 8$. This is shown in the following diagram.

The drone lands back on the platform when $t=p$.

Find the value of $p$.

The drone reaches its maximum height when $t=q$.

Find the value of $q$.

Find the drone's maximum height above the platform.

Find the drone's vertical distance from the platform when $t=8$.

The total vertical distance travelled by the drone in the first 8 seconds is given by $d$.

Find the value of $d$.

A second drone, Drone B, takes off from the same platform. Its velocity is given by $v_B(t) = 8 - 2t$, for $t \ge 0$.

When $t = k$, the total vertical distance travelled by Drone B is equal to $d$.

Find the value of $k$.

A mini-submarine is performing a test dive. Its velocity, $v$ in m/s, at time $t$ seconds is given by $v(t) = 5\sin(t) - t + 2$, for $0 \le t \le 8$.

(a) Calculate the time(s) when the submarine is momentarily at rest.

(b) Find the acceleration of the submarine when it first changes direction.

(c) Determine the total distance travelled by the submarine during the first 8 seconds of its dive.

A particle moves in a straight line. Its velocity, $v \,\text{ms}^{-1}$, at time $t$ seconds is given by $v(t) = -t^3 + 6t^2 - 9t$, for $0 \le t \le 5$. The particle is at the origin at $t=0$.

The graph of $v$ is shown in the following diagram.

(a) Find the displacement of the particle from the origin at $t=3$.

(b) Find an expression for the acceleration of the particle.

(c) The particle is momentarily at rest at $t=0$ and again at $t=k$. Find the greatest speed of the particle in the interval $0 \le t \le k$.

(d) Find the greatest speed of the particle for $0 \le t \le 5$.

(e) Write down an expression that represents the distance travelled by the particle while its speed is increasing. Do not evaluate the expression.

A drone is flying vertically. Its vertical velocity, $v\text{ms}^{-1}$, at time $t$ seconds is given by $v(t) = (t-4)\sin\left(\frac{t}{2}\right) + 1$, for $0 \leq t \leq 12$.

The following diagram shows the graph of $v$.

Find the smallest value of $t$ for which the drone is momentarily stationary.

Find the total vertical distance travelled by the drone during the first 12 seconds.

Find the acceleration of the drone when $t=7$ seconds.

A team of engineers is testing two autonomous robots, Alpha and Beta, on a straight track. Their positions are measured as the distance from a fixed starting point. The experiment runs for 10 minutes.

The position of Robot Alpha, $P_A$ metres, at time $t$ minutes can be modelled by the function $P_A(t) = 3\sin(2t + 5) + 14t + 20$, where $0 \le t \le 10$.

The position of Robot Beta, $P_B$ metres, at time $t$ minutes can be modelled by the function $P_B(t) = 12t + 25$, where $0 \le t \le 10$.

Use the engineers' models to find the initial position of

(i) Robot Beta;

(ii) Robot Alpha correct to three significant figures.

Find the values of $t$ when Robot Alpha and Robot Beta are at the same position. Give your answers correct to three significant figures.

For $t > 5$, prove that Robot Alpha was always ahead of Robot Beta.

For $0 \le t \le 10$, find the total amount of time when the speed of Robot Beta was greater than the speed of Robot Alpha. Give your answer correct to three significant figures.

A buoy floats on the surface of a lake. Its vertical displacement, $s$ metres, from its equilibrium position at time $t$ seconds is modelled by the function $s(t) = 5 \cos(2t+1)$, for $0 \leq t \leq 5$.

(a) Find the first time, $q$ seconds, when the buoy momentarily comes to rest.

(b) Find the total distance that the buoy travels in the first $q$ seconds.

A designer is creating a decorative glass container shaped like a dome. The outer profile of the container can be modelled by the function $f(x) = \sqrt{9-x^2}$, where $0 \le x \le 3$ and $x$ and $y$ are measured in metres.

Sketch the curve $y = f(x)$, clearly indicating the coordinates of the endpoints.

Show that the inverse function of $f$ is given by $f^{-1}(x) = \sqrt{9-x^2}$.

State the domain and range of $f^{-1}$.

The container is formed by rotating the curve $y = f(x)$ by $2\pi$ about the y-axis. Show that the volume, $V \text{ m}^3$, of liquid in the container when it is filled to a height of $h$ metres is given by $V = \pi \left( 9h - \frac{1}{3}h^3 \right)$.

Hence, determine the maximum volume of the container.

At $t = 0$, the container is empty. Liquid is then added to the container at a constant rate of $0.5 \text{ m}^3\text{s}^{-1}$.

Find the time it takes to fill the container to its maximum volume.

Find the rate of change of the height of the liquid when the container is filled to half its maximum volume.

A drone's altitude, $h$ metres, above the ground at time $t$ seconds is modelled by the function $h(t) = 8 \cos(0.5t + 1) + 12$, for $0 \leq t \leq 15$.

(a) Find the first time, $q$ seconds, when the drone's vertical velocity is zero.

(b) Find the total vertical distance that the drone travels in the first $q$ seconds.

A remote-controlled drone is launched from a platform and moves horizontally in a straight line. Its velocity, $v$ m s$^{-1}$, at time $t$ seconds after launch, is given by $v(t) = t + 1 - \sqrt{4t + 16}$, for $t \ge 0$.

(a) Find the drone's initial velocity.

(b) Find the time when the drone is at rest.

(c) Find the drone's acceleration at the instant it comes to rest.

(d) Determine the time interval(s) when the drone is:

(i) slowing down

(ii) speeding up.

A drone is programmed to ascend vertically from the ground. Its height, $h$ metres, $t$ seconds after launch, is modelled by the function $h(t) = 60t - 3t^2$.

(a) Find an expression for the vertical velocity, $v$, of the drone at time $t$.

(b) Calculate the maximum height reached by the drone.

The vertical displacement, $y$ metres, of a buoy from its equilibrium position in the water is modelled by the equation $y(t) = 8\sin^3(t)$, where $t$ is the time in seconds, $t \ge 0$.

(a) Find an expression for the acceleration of the buoy at time $t$.

(b) Find the values of $y$ at which the buoy is momentarily at rest.

(c) Find the values of $y$ for which the velocity of the buoy is at a maximum.

A hot air balloon takes off from a launch platform. Its height above the ground, $h$ metres, $t$ minutes after takeoff, is modelled by the function $h(t) = 24t - 0.8t^2$.

Determine the height of the hot air balloon after 5 minutes and after 10 minutes.

Find an expression for the vertical velocity of the hot air balloon, $v(t)$, in terms of $t$.

Calculate the time at which the hot air balloon reaches its maximum height.

Determine the total duration of the balloon's flight before it lands.

Sketch the graph of $h = h(t)$ for the duration of the flight, clearly indicating the intercepts with the $t$-axis and the coordinates of the maximum point. State the domain of the model.

Find the vertical velocity of the hot air balloon at the instant it lands.

Show that the vertical acceleration of the hot air balloon is constant, stating its value.

(a) A deep-sea submersible's vertical displacement, in metres, from a reference depth is given by $s(t) = 5 \sin(2t) - 3 \cos(t)$ for time $t$ minutes. Positive $s(t)$ indicates the submersible is above the reference depth, and negative $s(t)$ indicates it is below.

Find the submersible's vertical velocity and acceleration at any time $t$.

(b) Find the time intervals during $0 \le t \le 2\pi$ when the submersible is moving upwards.

(c) Determine the time intervals during $0 \le t \le 2\pi$ when the submersible's vertical velocity is decreasing.

(d) Calculate the total vertical distance travelled by the submersible during the time $0 \le t \le 2\pi$. Give your answer to three significant figures.

A drone takes off vertically from the ground. Its altitude, $h$ metres, above the ground at time $t$ seconds after takeoff is modelled by the function $h(t) = 50t - 2.5t^2$.

(a) Determine the altitude of the drone at $t=4$ seconds and $t=5$ seconds.

(b) Find an expression for the vertical velocity of the drone, $v(t)$, in terms of $t$.

(c) Find the instant in time, $t$, for which the drone reaches its maximum altitude.

(d) Determine how long it takes for the drone to return to the ground.

(e) Sketch the graph of $h = h(t)$, stating clearly the domain of validity of the model and its maximum point.

(f) Find the velocity of the drone at the instant it returns to the ground.

(g) Show that the acceleration of the drone is constant, stating its value.

A drone takes off vertically from the ground. Its velocity, $v$ in metres per second, is given by $v(t) = 3t^2 - 0.2t^3$ for $0 \le t \le 10$.

(a) Find an expression for the acceleration, $a$, of the drone at time $t$.

(b) Let $h$ be the height of the drone above the ground. Find an expression for $h$ in terms of $t$.