Variable Acceleration in 2D (Edexcel A Level Maths: Mechanics): Exam Questions

[In this question position vectors are given relative to a fixed origin $O$]

At time $t$ seconds, where $t\ge 0$, a particle, $P$, moves so that its velocity $\mathbf{v}$ ms^-1 is given by

$\mathbf{v}=6t\mathbf{i}-5t^{\frac{3}{2}}\mathbf{j}$

Find the acceleration of $P$ when $t=4$.

When $t=0$, the position vector of $P$ is $\left(-20\mathbf{i}+20\mathbf{j}\right)$ m.

Find the position vector of $P$ when $t=4$.

[In this question, position vectors are given relative to a fixed origin.]

At time $t$ seconds, where $t>0$, a particle $P$ has velocity $\mathbf{v}$ ms^–1 where

$\mathbf{v}=3t^2\mathbf{i}-6t^{\frac{1}{2}}\mathbf{j}$

Find the speed of $P$ at time $t=2$ seconds.

Find an expression, in terms of $t$, $\mathbf{i}$ and $\mathbf{j}$, for the acceleration of $P$ at time $t$ seconds, where $t>0$.

At time $t=4$ seconds, the position vector of $P$ is $\left(\mathbf{i}-4\mathbf{j}\right)$ m.

Find the position vector of $P$ at time $t=1$ second.

[In this question position vectors are given relative to a fixed origin $O$.]

At time $t$ seconds, where $t\ge 0$, a particle $P$ moves so that its velocity $\mathbf{v}{\mathrm{ms}}^{-1}$ is given by

$\mathbf{v}=\left(\begin{array}{c}{e}^t-t\\ 0.5t^4\end{array}\right)\mathrm{m}{\mathrm{s}}^{-1}$

When $t=0$, the particle $P$ is at the origin $O$.

Find the position vector of $P$ at time $t$ seconds.

Find the distance of the particle from $O$ at time $t=1\mathrm{s}$, giving your answer to three significant figures.

[In this question, position vectors are given relative to a fixed origin $O$.]

The velocity $\mathbf{v}$ of a particle at time $t$ seconds, where $t\ge 0$ is given by

$\mathbf{v}=\left(\left(10t-3t^2\right)\mathbf{i}+\left(4t-5\right)\mathbf{j}\right)\mathrm{m}{\mathrm{s}}^{-1}$

Find the initial acceleration of the particle.

The particle’s initial position is at the point $\left(4\mathbf{i}+5\mathbf{j}\right)\mathrm{m}$.

Find the distance of the particle from the origin at time $t=3$seconds, giving your answer to three significant figures.

[In this question, position vectors are given relative to a fixed origin $O$.]

The position vector of a boat, sailing on a lake is

 $\mathbf{r}=\left(2\sin t\right)\mathbf{i}+(2-2\cos t)\mathbf{j}\mathrm{km}$ 

where time $t$ is measured in hours and $t\ge 0$.

Show that the boat takes $2\pi$ hours until it first returns to $O$.

Find the velocity of the boat at time $t=\frac{2}{3}\mathrm{\pi }$ hours.

[In this question, position vectors are given relative to a fixed origin $O$.]

A particle $P$ moves in a plane with velocity, $\mathbf{v}\mathrm{m}{\mathrm{s}}^{-1}$, at time $t$ seconds where $t\ge 0$. $\mathbf{v}$ is given by

$\mathbf{v}=\left(\begin{array}{c}4t-3t^2\\ 6t^2-2\end{array}\right)$ 

Find the acceleration of the particle at time  $t=3$ seconds.

Given that its initial position is at the origin, find the position vector of the particle at time $t=4$ seconds.

[In this question $\mathbf{i}$ and $\mathbf{j}$ are horizontal unit vectors.]

An aircraft is modelled as a particle $P$ moving in a horizontal plane. At time $t$ seconds, where $t\ge 0$, the acceleration of $P$, $\mathbf{a}\mathrm{m}{\mathrm{s}}^{-2}$, is given by

$\mathbf{a}=(30-2t)\mathbf{i}+(4t-3)\mathbf{j}$

When $t=0$, the velocity of $P$ is $(200\mathbf{i}+150\mathbf{j})\mathrm{m}{\mathrm{s}}^{-1}$.

Find the velocity of $P$ at time $t$ seconds.

Find the speed of $P$ at time $t=4$ seconds, giving your answer in kilometres per hour to three significant figures.

A remote-controlled car is driven around a playground with velocity, $\mathbf{v}\mathrm{m}{\mathrm{s}}^{-1}$, at time $t$ seconds, given by

 $\mathbf{v}=(0.25)\mathbf{i}+(0.5t-9)\mathbf{j}$ 

Find an expression for the displacement of the remote-controlled car, $\mathbf{s}$ metres, measured from its initial position.

The remote-controlled car starts from the point $\left(2\mathbf{i}-5\mathbf{j}\right)\mathrm{m}$ metres.

Find the position vector $\mathbf{r}$ of the particle from the origin at time $t$ seconds.

Find the distance of the remote-controlled car from the origin after 40 seconds.

[In this question $\mathbf{i}$ and $\mathbf{j}$ are horizontal unit vectors and position vectors are given relative to a fixed origin $O$.]

A spider is modelled as a particle $P$ moving on a horizontal floor. At time $t$ seconds, where $t\ge 0$, the acceleration of $P$, $\mathbf{a}\mathrm{m}{\mathrm{s}}^{-2}$, is given by

$\mathbf{a}=(0.1t)\mathbf{i}+(0.6t^2-2t)\mathbf{j}\mathrm{m}{\mathrm{s}}^{-2}$

The velocity of $P$ at time $t=0$ is $(-1.2\mathbf{i}+1.8\mathbf{j})\mathrm{m}{\mathrm{s}}^{-1}$.

Find the speed of $P$ at time $t=3$.

Find the position vector of $P$ at time $t$ seconds.

[In this question, position vectors are given relative to a fixed origin $O$.]

At time $t$ seconds, where $t\ge 0$, a particle $P$ moves so that its position vector $\mathbf{r}$ metres is given by

 $\mathbf{r}=\left(12e^{-0.1t}\right)\mathbf{i}+\left(24e^{-0.2t}\right)\mathbf{j}\mathrm{m}$

Explain why $P$ never reaches the origin $O$.

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

Find the magnitude of acceleration of $P$ at time $t=4$ seconds, giving your answer to three significant figures.

A particle’s velocity is modelled by the equation

 $\stackrel{\mathbf{·}}{\mathbf{r}}=\left(\begin{array}{c}3t^2-6t\\ 4-8t^3\end{array}\right)\mathrm{m}{\mathrm{s}}^{-1}$

where $t$ is the time in seconds.

Given that the particle is initially located at the point $\left(\begin{array}{c}2\\ 1\end{array}\right)\mathrm{m}$, find the position vector of the particle, $\mathbf{r}\mathrm{m}$, at time $t$ seconds.

Find the time at which the particle has zero acceleration in the horizontal direction.

[In this question, position vectors are given relative to a fixed origin $O$.]

At time $t$ seconds, where $t\ge 0$, a particle $P$ moves so that its position vector $\mathbf{r}$ metres is given by

$\mathbf{r}=\left(t^3-11t^2-16t+2\right)\mathbf{i}+\left(t^3+2t-1\right)\mathbf{j}$

Find the velocity of $P$ at time $t$ seconds.

Find the value of $t$ at the instant when $P$ is moving in a direction parallel to $\mathbf{j}$.

Show that $P$ never moves in a direction perpendicular to $\mathbf{j}$.

[In this question $\mathbf{i}$ and $\mathbf{j}$ are horizontal unit vectors and position vectors are given relative to a fixed origin $O$.]

At time $t$ seconds, where $t\ge 0$, a particle $P$ is moving in a horizontal plane with velocity $\mathbf{v}\mathrm{m}{\mathrm{s}}^{-1}$ given by

$\mathbf{v}=(0.1t^3-3t^2)\mathbf{i}+(2t+1)\mathbf{j}$

Find the acceleration of $P$ at time $t$ seconds.

Explain why the acceleration of $P$ in the direction of $\mathbf{j}$ is constant.

Find the value of $t$, other than $t=0$, for which the acceleration of $P$ in the direction of $\mathbf{i}$ is zero.

When $t=0$, $P$ is at the point with position vector $(-3\mathbf{i}+5\mathbf{j})\mathrm{m}$.

Find the position vector of $P$ at time $t$ seconds.

[In this question $\mathbf{i}$ and $\mathbf{j}$ are horizontal unit vectors and position vectors are given relative to a fixed origin $O$.]

An ice skater is modelled as a particle $P$ moving across a straight section of a frozen river. At time $t$ seconds, where $t\ge 0$, the position vector of $P$, $\mathbf{r}$ metres, is given by

$\mathbf{r}=\left(\frac{1}{3}{t}^2+\frac{1}{5}t\right)\mathbf{i}+\left(2t^2+7t\right)\mathbf{j}$

Find the speed of $P$ at time $t=0$, giving your answer to 3 significant figures.

Show that the acceleration of $P$ is constant and find the magnitude of this acceleration, giving your answer to 3 significant figures.

At time $t$ seconds, where $t\ge 0$, a particle $P$ moves in the $x$-$y$ plane in such a way that its velocity $v$ms⁻¹ is given by

$\mathbf{v}=t^{-\frac{1}{2}}\mathbf{i}-4t\mathbf{j}$

When $t=1$, $P$ is at the point $A$ and when $t=4$, $P$ is at the point $B$.

Find the exact distance $AB$.

At time $t$ seconds, where $t\ge 0$, a particle $P$ moves so that its acceleration $\mathbf{a}$ ms⁻² is given by

$\mathbf{a}=\left(1-4t\right)\mathbf{i}+\left(3-t^2\right)\mathbf{j}$

At the instant when $t=0$, the velocity of $P$ is $36\mathbf{i}$ ms^-1.

Find the velocity of $P$ when $t=4$.

Find the value of $t$ at the instant when $P$ is moving in a direction perpendicular to $\mathbf{i}$.

At time $t$ seconds, where $t\ge 0$ , a particle $Q$ moves so that its position vector $\mathbf{r}$ metres, relative to a fixed origin $O$, is given by

$\mathbf{r}=\left(t^2-t\right)\mathbf{i}+3t\mathbf{j}$

Find the value of $t$ at the instant when the speed of $Q$ is 5 ms⁻¹.

At time $t$ seconds, where $t\ge 0$, a particle $P$ has velocity $\mathbf{v}$ ms^–1 where

$\mathbf{v}=\left(t^2-3t+7\right)\mathbf{i}+\left(2t^2-3\right)\mathbf{j}$

Find the speed of $P$ at time $t=0$.

Find the value of $t$ when $P$ is moving parallel to $\left(\mathbf{i}+\mathbf{j}\right)$.

Find the acceleration of $P$ at time $t$ seconds.

Find the value of $t$ when the direction of the acceleration of $P$ is perpendicular to $\mathbf{i}$.

At time $t$ seconds, a particle $P$ has velocity $\mathbf{v}$ ms⁻¹, where

$\mathbf{v}=3t^{\frac{1}{2}}\mathbf{i}-2t\mathbf{j}t>0$

Find the acceleration of $P$ at time $t$ seconds, where $t>0$.

Find the value of $t$ at the instant when $P$ is moving in the direction of $\mathbf{i}-\mathbf{j}$.

At time $t$ seconds, where $t>0$, the position vector of $P$, relative to a fixed origin $O$, is $\mathbf{r}$ metres.

When $t=1$, $\mathbf{r}=-\mathbf{j}$.

Find an expression for $\mathbf{r}$ in terms of $t$.

Find the exact distance of $P$ from $O$ at the instant when $P$ is moving with speed 10 ms⁻¹.

[In this question $\mathbf{i}$ and $\mathbf{j}$ are horizontal unit vectors and position vectors are given relative to a fixed origin $O$.]

A boat is modelled as a particle $B$ moving on a lake. At time $t$ seconds, where $t\ge 0$, the position vector of $B$, $\mathbf{r}$ metres, is given by

$\mathbf{r}=(-20\sin (\frac{t}{360}\left)\right)\mathbf{i}+(20-20\cos (\frac{t}{360}\left)\right)\mathbf{j}\mathrm{m}$

Show that the distance of $B$ from $O$ when $t=180\pi$ is $20\sqrt{2}\mathrm{m}$.

Show that the speed of $B$ at the instant it first returns to $O$ is $\frac{1}{18}\mathrm{m}{\mathrm{s}}^{-1}$.

[In this question $\mathbf{i}$ and $\mathbf{j}$ are horizontal unit vectors.]

An aircraft is modelled as a particle $P$ moving in a horizontal plane. Once $P$ reaches its cruising height, at time $t$ hours where $t\ge 0$, the acceleration of $P$, $\mathbf{a}\mathrm{km}{\mathrm{h}}^{-2}$, is given by

 $\mathbf{a}=\left(4t^3-6t^2\right)\mathbf{i}+\left(0.9t^2-1\right)\mathbf{j}$

The velocity of $P$ at time $t=5$ is $(400\mathbf{i}+40\mathbf{j})\mathrm{km}{\mathrm{h}}^{-1}$.

Find the velocity of $P$ at time $t$ hours.

Find the speed of $P$ when $t=0$.

[In this question $\mathbf{i}$ and $\mathbf{j}$ are horizontal unit vectors and position vectors are given relative to a fixed origin $O$.]

A remote-controlled car is modelled as a particle $C$ moving in a horizontal plane. At time $t$ seconds, where $t\ge 0$, the velocity of $C$, ${\mathbf{v}}_C\mathrm{m}{\mathrm{s}}^{-1}$, is given by

${\mathbf{v}}_{\mathit{c}}=\left(0.45{\mathrm{t}}^2+2\mathrm{t}-16\right)\mathbf{i}+\left(0.75{\mathrm{t}}^2-1\right)\mathbf{j}$

When $t=0$, the position vector of $C$ is $(-6\mathbf{i}+15\mathbf{j})\mathrm{m}$.

(i) Find the position vector of $C$ at time $t$ seconds.

(ii) Find the distance of $C$ from $O$ at the instant when $t=15$.

At the same time as the car starts moving, a remote-controlled truck, modelled as a particle $T$, is set in motion. The position vector of $T$, ${\mathbf{r}}_T$​ metres, at time $t$ seconds is given by

${\mathbf{r}}_T=(0.15t^3-6)\mathbf{i}+(0.25t^3-1)\mathbf{j}$

Determine the time at which the car and the truck collide.

[In this question $\mathbf{i}$ and $\mathbf{j}$ are horizontal unit vectors and position vectors are given relative to a fixed origin $O$.]

A spider is modelled as a particle $P$ moving on a horizontal floor. At time $t$ seconds after emerging from under a skirting board, where $t\ge 0$, the acceleration of $P$, $\mathbf{a}\mathrm{m}{\mathrm{s}}^{-2}$, is given by

$\mathbf{a}=\left(1.2t\right)\mathbf{i}+0.5\mathbf{j}$

When $t=3$, the velocity of $P$ is $(5.4\mathbf{i}+1.7\mathbf{j})\mathrm{m}{\mathrm{s}}^{-1}$.

Find the velocity of $P$ at time $t$ seconds.

When $t=3$, the position vector of $P$ is $(10.4\mathbf{i}+5.15\mathbf{j})\mathrm{m}$.

Find the distance of $P$ from $O$ at the instant it emerges from under the skirting board.

[In this question $\mathbf{i}$ is a horizontal unit vector and $\mathbf{j}$ is a vertical unit vector directed upwards. Position vectors are given relative to a fixed origin $O$.]

A stone is modelled as a particle $P$ thrown from a point $O$ on the edge of a deep hole. At time $t$ seconds after being thrown, where $t\ge 0$, the velocity of $P$, $\mathbf{v}\mathrm{m}{\mathrm{s}}^{-1}$, is given by

$\mathbf{v}=\left(0.2t\right)\mathbf{i}+\left(4-0.75{\mathrm{t}}^2\right)\mathbf{j}$

Find the position vector of $P$ at time $t=4$ and give an interpretation of the $\mathbf{j}$ component of this position vector.

The hole has a depth of 384 m.

Find the magnitude of the acceleration of $P$ at the instant it hits the bottom of the hole.

A particle’s velocity is modelled by the equation

$\stackrel{\mathbf{·}}{\mathbf{r}}=\left(\begin{array}{c}0.75e^{1.5t}+2t\\ 5t-{\left(t+1\right)}^{-1}\end{array}\right)\mathrm{m}{\mathrm{s}}^{-1}t\ge 0$

where $t$ is the time in seconds.

The particle’s initial displacement is $\left(\begin{array}{c}0\\ 0\end{array}\right)\mathrm{m}$.

Find an expression for $\mathbf{r}$, at time $t$ seconds. State appropriate units with your answer.

Find $\left|\stackrel{\mathbf{·}\mathbf{·}}{\mathbf{r}}\right|$ when $t=1\mathrm{s}$. State appropriate units with your answer.

[In this question $\mathbf{i}$ and $\mathbf{j}$ are horizontal unit vectors and position vectors are given relative to a fixed origin $O$.]

At time $t$ seconds, where $t\ge 0$, a particle $P$ moves so that its acceleration, $\mathbf{a}\mathrm{m}{\mathrm{s}}^{-2}$, is given by

$\mathbf{a}=(4\mathrm{t}-3)\mathbf{i}+(4\mathrm{t}+5)\mathbf{j}$

When $t=0$, the velocity of $P$ is $-5\mathbf{j}\mathrm{m}{\mathrm{s}}^{-1}$ and $P$ is at the origin $O$.

Find the distance of $P$ from $O$ when $t=6$.

Find the value of $t$ at the instant when $P$ is moving in the direction of $(\mathbf{i}+2\mathbf{j})$.

[In this question $\mathbf{i}$ and $\mathbf{j}$ are horizontal unit vectors and position vectors are given relative to a fixed origin $O$ on one bank of a straight river.]

An ice skater is modelled as a particle $P$ moving across the river. The banks of the river are parallel to the vector $\mathbf{i}$. At time $t$ seconds, where $0\le t\le 225$, the position vector of $P$, $\mathbf{r}$ metres, is given by

$\mathbf{r}=(2t^{\frac{1}{2}}+t)\mathbf{i}+t^{\frac{1}{2}}\mathbf{j}$

Find the distance of $P$ from $O$ when $t=25$.

Given that the skater reaches the opposite bank of the river when $t=225$, find the width of the river.

Show that the magnitude of the acceleration of $P$ at time $t$ seconds is given by $\left|\mathbf{a}\right|=0.25\sqrt{5t^{-3}}\mathrm{m}{\mathrm{s}}^{-2}$.