Calculus for Kinematics (Cambridge (CIE) IGCSE Additional Maths): Exam Questions

In this question, all lengths are in metres and time, $t$, is in seconds.

The diagram shows the displacement–time graph for a runner, for $0⩽t⩽40$.

(i) Find the distance the runner has travelled when $t=40$.

(ii) On the axes, draw the corresponding velocity–time graph for the runner, for $0⩽t⩽40.$

A particle travels in a straight line. As it passes through a fixed point $O$, the particle is travelling at a velocity of $3{\mathrm{ms}}^{–1}$. The particle continues at this velocity for 60 seconds then decelerates at a constant rate for 15 seconds to a velocity of $1.6{\mathrm{ms}}^{–1}$. The particle then decelerates again at a constant rate for 5 seconds to reach point $A$, where it stops.

Sketch the velocity-time graph for this journey on the axes below.

Find the distance between $O$ and $A$.

Find the deceleration in the last 5 seconds.

The diagram shows the $x–t$ graph for a runner, where displacement, $x$, is measured in metres and time, $t$, is measured in seconds.

(i) On the axes below, draw the $v–t$ graph for the runner.

[3]

(ii) Find the total distance covered by the runner in 125 s.

[1]

A particle $P$ moves in a straight line such that its displacement, $x\mathrm{m}$, from a fixed point $O$ at time $t\mathrm{s}$ is given by  $x=10\sin 2t-5$.

(i) Find the speed of $P$ when $t=\mathrm{\pi }$.

[1]

(ii) Find the value of $t$ for which $P$ is first at rest.

[2]

(iii) Find the acceleration of $P$ when it is first at rest.

[2]

In this question, the units are metres and seconds. A particle $P$ is travelling in a straight line. Its acceleration, $a$, away from a fixed point $O$, at time $t$, is given by $a={\left(3t+2\right)}^{-\frac{1}{3}}$, where $t⩾0$. When$t=2$, $P$ is travelling with a velocity of 8 and has a displacement of $–4.8$ from $O$.

Find an expression for the velocity of $P$ at time $t$.

Explain why $P$ is never at rest.

Find the displacement of $P$ from $O$ when $t=\frac{25}{3}$ .

A particle, $P$, moves in a straight line such that its displacement from a fixed point at time $t$ is s.

The acceleration of $P$ is given by ${\left(2t+4\right)}^{-\frac{1}{2}},\mathrm{for}t>0$

(i) Given that $P$ has a velocity of 9 when $t=6$, find the velocity of $P$ at time $t$.

(ii) Given that s = $\frac{1}{3}$ when $t=6$, find the displacement of $P$ at time $t$.

At time $t\mathrm{s}$, a particle travelling in a straight line has acceleration ${\left(2t+1\right)}^{-\frac{1}{2}}{\mathrm{ms}}^{-2}$. When $t=0$, the particle is $4\mathrm{m}$ from a fixed point $O$ and is travelling with velocity $8{\mathrm{ms}}^{-1}$ away from $O$.

Find the velocity of the particle at time $t\mathrm{s}$.

Find the displacement of the particle from $O$ at time $t\mathrm{s}$.

A particle travels along a straight line with a velocity, $v$ ms^-1, given by

$v=6{\left(t-\frac{1}{2}\right)}^2+\frac{5}{2}$

for $t\ge 0$ where $t$ is time in seconds. The particle has an initial displacement of $-1$ metres.

Find the acceleration after 2 seconds.

Find the displacement after 2 seconds.

Explain why a velocity-time graph and a speed-time graph will always be the same for this particle.

A particle moves in a straight line such that, $t$ seconds after passing a fixed point $O$, its displacement from $O$ is $s\mathrm{m}$, where $s={\mathrm{e}}^{2t}-10{\mathrm{e}}^t-12t+9$.

Find expressions for the velocity and acceleration at time $t$.

Find the time when the particle is instantaneously at rest.

Find the acceleration at this time.