Kinematics (DP IB Applications & Interpretation (AI): HL): Exam Questions

A cricket ball is projected vertically upwards from ground level. Its height above the ground, $h$ metres, $t$ seconds after it is projected can be modelled by $h\left(t\right)=13t-4.9t^2$, for $t\ge 0$.

Find the times at which the ball is 3 m above the ground.

Find the length of time for which the ball is at least 3 m above the ground.

A player catches the ball on its way down, when it is 0.8 m above the ground.

Find the length of time that the ball is in the air.

Find the total distance travelled by the ball before it is caught.

Find the velocity of the ball when $t=1$.

A particle moves in a straight line such that its velocity, $v$ m s^-1, at time $t$ seconds is given by $v\left(t\right)=2^t-2$, for $0\le t\le 4$.

Find the acceleration of the particle when $t=2$.

Find the value of $t$ when the particle is at rest.

Find the total distance travelled by the particle.

A particle moves in a straight line. Its acceleration, $a$ m s^-2, at time $t$ seconds is given by $a\left(t\right)=\frac{1}{t^2}+\sin t$, for $t\ge 1$.

When $t=1$, the velocity of the particle is 1 m s^-1.

Find an expression for the velocity, $v$ m s^-1, of the particle at time $t$.

Find the velocity of the particle when $t=2$.

A particle moves in a straight line such that its velocity, $v$ m s^-1, at time $t$ seconds is given by $v\left(t\right)=\sin t+\cos 2t$, for $0\le t\le 5$.

Find the values of $t$ when the particle is at rest.

Hence find the value of $t$ at which the particle changes direction.

Find the distance travelled by the particle in the first second of its motion.

Find the acceleration of the particle at the instant it changes direction.

Find the displacement of the particle from its starting point when $t=5$.

A particle is moving along a straight line.  The position of the particle at time $t$ seconds, measured in metres relative to a fixed origin point, is denoted by  $x\left(t\right)$.

The particle starts at the origin at time $t=0$ ,  and its motion over the next eight seconds is described by the equation

$x\left(t\right)=\frac{1}{{\cos }^2\left({\displaystyle \frac{\mathrm{\pi }}{20}}t\right)}-3,0\le t\le 8$

Find an expression for $x\left(t\right)$.

Hence determine the maximum distance of the particle from the origin during the first eight seconds of its movement.

Find the change in displacement of the particle during the first eight seconds of its movement.

Find the total distance travelled by the particle during the first eight seconds of its movement.

Find an expression for the particle’s acceleration $\ddot{x}\left(t\right)$.

A particle is moving along a straight line.  The position of the particle at any given time, measured in metres relative to a fixed origin point, is denoted by $x$.

It is known that the velocity,$v{\mathrm{ms}}^{-1}$ , of the particle is dependent on the particle’s position, and that the velocity may be described by the equation

$v\left(x\right)=\sqrt{1-x^2},-1\le x\le 1$

Use the chain rule to explain why the acceleration, $a{\mathrm{ms}}^{-2}$, of the particle may be expressed in the form 

$a=v\frac{dv}{dx}$

Show that the derivative of  $\sqrt{1-x^2}$  is $-\frac{x}{\sqrt{1-x^2}}$ .

Hence find an expression for the acceleration of the particle in terms of $x$, being sure to indicate the domain of $x$ values for which the expression is valid.

A marble is projected along a marble run that lies in a vertical plane through a fixed point $\mathrm{O}$. The height of the marble above $\mathrm{O}$, $h$ centimetres, $t$ seconds after it is projected can be modelled by $h\left(t\right)=15-10t+2t^2$, for $0\le t\le 6$.

Write down the initial height of the marble above $\mathrm{O}$.

Find the time at which the marble reaches its lowest point, and the total vertical distance travelled by the marble up to that time.

Find the velocity and the acceleration of the marble in the vertical direction when $t=6$.

A particle moves in a straight line. Its displacement, $s$ metres, from a fixed point $\mathrm{P}$ at time $t$ seconds is given by $s\left(t\right)=\frac{t^3}{20}-\frac{17t^2}{8}+18t-4$, for $0\le t\le 30$.

Write down the initial distance of the particle from $\mathrm{P}$.

Find the displacement of the particle from $\mathrm{P}$ when $t=30$.

Find an expression for the velocity, $v$ m s^-1, of the particle at time $t$.

Find the values of $t$ when the particle is at rest.

A particle moves in a straight line such that its velocity, $v$ m s^-1, at time $t$ seconds is given by $v\left(t\right)=t^3\cos t$, for $t\ge 0$.

Find an expression for the acceleration, $a$ m s^-2, of the particle at time $t$.

Hence find the acceleration of the particle when $t=5.2$.

A particle moves in a straight line such that its velocity, $v$ m s^-1, at time $t$ seconds is given by $v\left(t\right)=\sqrt{t^2+2t}-3$, for $0\le t\le 5$.

Find the acceleration of the particle when $t=3.7$.

(i) Find the displacement of the particle from its starting position when $t=1$.

(ii) Explain what your answer to part (b)(i) tells you about the motion of the particle for $0\le t\le 1$.

Find the total distance travelled by the particle.

A ball is placed at the midpoint, X, of a horizontal tube of length 120 cm. A player at each end of the tube pumps air into it to move the ball.

The following graph shows the velocity, $v$ cm s^-1, of the ball at time $t$ seconds during a 20-second game.

Write down the maximum speed of the ball and the interval of time during which it occurs.

There are two intervals of time during which the ball is decelerating. Find the magnitude of the deceleration of the ball after it first changes direction.

Find the total distance travelled by the ball during the game.

A particle moves in a straight line. Its displacement, $s$ metres, from a fixed point $\mathrm{O}$ at time $t$ seconds is given by $s\left(t\right)=\frac{1}{2}t-\sin 2t$, for $0\le t\le 6$.

Write down the number of times the particle changes direction.

Find an expression for the velocity, $v$ m s^-1, of the particle at time $t$.

Find the maximum velocity of the particle and the values of $t$ at which it occurs.

Find the percentage of the total distance travelled by the particle that is travelled in the first 2 seconds.

A particle is moving along a straight line.  The position of the particle at time  seconds, measured in metres relative to a fixed origin point, is denoted by $x\left(t\right)$ .

The particle starts at rest at the origin at time $t=0$ ,  and its motion over the next six seconds is described by the equation

$\ddot{x}\left(t\right)=\frac{4{\pi }^2}{9}\sin \left(\frac{2\pi t}{3}\right)-\frac{{\mathrm{\pi }}^2}{36}\cos \left(\frac{\pi t}{12}\right),0\le t\le 6$

Find expressions for

(i)       $\dot{x}\left(t\right)$

(ii)      $x\left(t\right)$.

Find the maximum value that each of the following quantities takes on during the first six seconds of movement, as well as the time $t$ at which those maximum values occur:

(i)      the distance of the particle from the origin

(ii)     the speed of the particle

(iii)    the magnitude of the particle’s acceleration.

Find the total distance travelled by the particle during the first six seconds of its movement.

A particle is moving along a straight line.  The position of the particle at any given time, measured in metres relative to a fixed origin point, is denoted by $x$.

It is known that the velocity, $v{\mathrm{ms}}^{-1}$, of the particle is dependent on the particle’s position, and that the velocity may be described by the function

$v\left(x\right)=-\sqrt{4-9x^2},-\frac{2}{3}\le x\le \frac{2}{3}$

Show that the acceleration, $a{\mathrm{ms}}^{-2}$, of the particle may be expressed in the form

$a=v\frac{dv}{dx}$

Hence find a function giving the acceleration of the particle in terms of $x$.

Identify the minimum and maximum values of

          (i)     the speed of the particle

          (ii)     the magnitude of the particle’s acceleration

along with the values of $x$ for which those occur.

A particle P moves in a straight line. Its velocity, $v_{\mathrm{P}}$ m s^-1, at time $t$ seconds is given by

$v_{\mathrm{P}}\left(t\right)=t^2\cos \left(\frac{\pi }{4}t\right)$

for $0\le t\le 10$.

Write down the value of $t$ at which P first changes direction.

Find the total distance travelled by P while its speed is increasing.

A second particle, Q, also moves in a straight line. Its velocity, $v_{\mathrm{Q}}$ m s^-1, at time $t$ seconds is given by

$v_{\mathrm{Q}}\left(t\right)=6t+2$

for $0\le t\le 8$.

After $k$ seconds, the total distance travelled by Q is equal to the distance travelled by P while its speed is increasing.

Find the value of $k$.

The height, $h$ metres, of the water level in an estuary relative to the mean sea level, $t$ hours after midnight, can be modelled by

$h\left(t\right)=\sin \left(0.262t-0.5\right)-3\cos \left(0.524t\right)+1$

for $0\le t\le 24$.

Find the rate of change of the height of the water level at 7 am.

Find the percentage of the 24-hour period for which the water level is above the mean sea level.

A buoy is anchored in the estuary so that its horizontal position is fixed, but it moves up and down with the water level.

Find the total vertical distance travelled by the buoy during the 24-hour period.

A particle moves in a straight line, starting from a fixed point $\mathrm{P}$. Its velocity, $v$ m s^-1, at time $t$ seconds is given by

$v\left(t\right)=\{\begin{array}{cc}9t-3t^2,& 0\le t\le 4\\ -3t+\frac{16}{t^3}-\frac{1}{4},& 4<t\le 10\end{array}$

Find the maximum velocity of the particle and the value of $t$ at which it occurs.

Find an expression for the displacement, $s$ metres, of the particle from $\mathrm{P}$ at time $t$, given that $s\left(10\right)=-119.08$.

Find the total distance travelled by the particle.

A toy car moves along a straight horizontal track, starting from a fixed point $\mathrm{P}$. Its displacement, $s$ centimetres, from $\mathrm{P}$ at time $t$ seconds can be modelled by

$s\left(t\right)=\frac{1}{1000}t\left(t^2-190t+8400\right)$

for $0\le t\le 140$.

Sketch the graph of $s$ against $t$.

Find an expression for the acceleration, $a$ cm s^-2, of the toy car at time $t$. Hence find the acceleration of the toy car when $t=85$.

Find the greatest speed of the toy car while its displacement is decreasing.

A particle starts from a fixed point $\mathrm{X}$ and moves in a straight line. The following graph shows its velocity, $v$ m s^-1, at time $t$ seconds, for $0\le t\le 8$.

The particle is instantaneously at rest when $t=0$, $t=3$, $t=5$ and $t=8$.

Let $s\left(t\right)$ metres be the displacement of the particle from $\mathrm{X}$ at time $t$.

The particle travels 22 metres in the first 3 seconds. Also, $s\left(3\right)=s\left(7\right)$ and ${\int }_3^{5}v\mathrm{d}t=9$.

Write down the value of $s\left(5\right)-s\left(3\right)$.

Find the total distance travelled by the particle in the first 7 seconds.

A particle A moves along a horizontal straight line $L_1$. Its displacement, $s_{\mathrm{A}}$ metres, from a fixed point $\mathrm{P}$ on $L_1$ at time $t$ seconds is given by

$s_{\mathrm{A}}\left(t\right)=\frac{1}{2}t-2t^3{\mathrm{e}}^{-0.3t}+24$

for $0\le t\le 22$.

At the same time, a second particle, B, starts to move along a horizontal straight line $L_2$, parallel to $L_1$. Its velocity, $v_{\mathrm{B}}$ m s^-1, at time $t$ seconds is given by

$v_{\mathrm{B}}\left(t\right)=\frac{1}{2}t+12$

for $0\le t\le 22$.

Find the values of $t$ for which particle A is at $\mathrm{P}$.

Find the value of $t$ at which particle A first changes direction.

Find the total distance travelled by particle A in the first 8 seconds of its motion.

The displacement, $s_{\mathrm{B}}$ metres, of particle B is measured from a fixed point $\mathrm{Q}$ on $L_2$. It is given that $s_{\mathrm{A}}\left(0\right)=s_{\mathrm{B}}\left(5\right)$.

(i) Find an expression for $s_{\mathrm{B}}\left(t\right)$.

(ii) Find the displacement of each particle at the time when $s_{\mathrm{A}}\left(t\right)=s_{\mathrm{B}}\left(t\right)$.

Two particles, ${\mathrm{P}}_1$ and ${\mathrm{P}}_2$, move along the same straight line. Their displacements, $s_1$ metres and $s_2$ metres, from a fixed point $\mathrm{O}$ at time $t$ seconds are given by

$s_1\left(t\right)=\frac{1}{2}\sin \left(t-0.9\right)-\cos \left(2t-1.8\right)-1$

$s_2\left(t\right)=\cos \left(6t-5.4\right)-\sin \left(t-0.9\right)+2.5$

for $0\le t\le 3$.

Find an expression for the distance between the two particles at time $t$.

Hence find

(i) the maximum distance between the particles

(ii) the value of $t$ at which this maximum distance occurs.

The particles collide once during this time.

Find the velocity of each particle 0.5 seconds before they collide.

A particle is moving along a straight line.  The position of the particle at time $t$ seconds, measured in metres relative to a fixed origin point, is denoted by $x\left(t\right)$ .

The particle starts at the origin at time  $t=0$  with a velocity of $-3{\mathrm{ms}}^{-1}$ ,  and its motion over the next ten seconds is described by the equation

$\ddot{x}\left(t\right)=\frac{3}{32}{e}^{\frac{3t}{8}}-\frac{9{\pi }^2}{25}cos\frac{3\pi t}{5},0\le t\le 10$

Considering the total distance travelled by the particle over the ten seconds, calculate the percentage of that total distance that the particle travels in the first five seconds of its movement.

Find the greatest distance from the origin point that the particle reaches, and the time $t$  at which that greatest distance is reached.

Professor Goodwin Vundera, a kinematics researcher, has been studying a particular type of particle.  The particle is known only to move along a straight line, with its acceleration,$a\mathrm{m}{\mathrm{s}}^{-2}$ , and velocity,$v{\mathrm{ms}}^{-1}$ , both dependent on the particle’s displacement, $x$ m, with reference to a fixed origin point.

The professor has defined a new real-valued function, the ‘Vundera function’, which he believes captures all necessary information about the motion of the particle.  This Vundera function, $W$, is defined by  $W\left(x\right)=\frac{a\left(x\right)}{v\left(x\right)}$ ,  where  $a\left(x\right)$ and $v\left(x\right)$ are functions describing, respectively, the particle’s acceleration and velocity in terms of $x$.  For one of the particles studied by the professor, the associated Vundera function is found to be

$W\left(x\right)=\frac{2x}{\sqrt{225-9x^2}},a<x<b$

where $a$ and $b$ are real constants.

Given that $W$ has the largest possible domain, write down the values of $a$ and $b$.

Additionally it is known that when $x=4$, the velocity of the particle is $-2{\mathrm{ms}}^{-1}$.

By using the above information and solving an appropriate indefinite integral, find expressions for the functions $v\left(x\right)$ and $a\left(x\right)$ associated with the particle.

Explain the relationship between the particle’s speed and acceleration as $x$ varies across all the values in the domain.