A particle moves in a straight line, starting at time $t=0$ and continuing until it comes to rest. While it is moving the particle has acceleration $a \mathrm{m}{ \mathrm{s}}^{-2}$, in the positive direction along the line. This acceleration is given by $a=0.1-0.01t$, where $t$ is the time from the start, in seconds. The particle starts with speed $4 \mathrm{m}{ \mathrm{s}}^{-1}$ and finishes with speed $0 \mathrm{m}{ \mathrm{s}}^{-1}$. Find the time when the particle comes to rest.

A car is moving in a straight line. The acceleration, $a \mathrm{m}{ \mathrm{s}}^{-2}$, at time $t \mathrm{s}$ after the car starts to move is modelled as $a=A(1+4t)$ for $0⩽t<1$ and $a=B\left(30-\frac{10}{t^3}\right)$ for $1⩽t⩽5$, where $A$ and $B$ are constants.

Show that $A=4B$

A car is moving in a straight line. The acceleration, $a \mathrm{m}{ \mathrm{s}}^{-2}$, at time $t \mathrm{s}$ after the car starts to move is modelled as $a=A(1+4t)$ for $0⩽t<1$ and $a=B\left(30-\frac{10}{t^3}\right)$ for $1⩽t⩽5$, where $A$ and $B$ are constants. At $t=5 \mathrm{s}$ the velocity of the car is $31.8 \mathrm{m}{ \mathrm{s}}^{-1}$. Show that $A=1$.

A car is moving in a straight line. The acceleration, $a \mathrm{m}{ \mathrm{s}}^{-2}$, at time $t \mathrm{s}$ after the car starts to move is modelled as $a=A(1+4t)$ for $0⩽t<1$ and $a=B\left(30-\frac{10}{t^3}\right)$ for $1⩽t⩽5$, where $A$ and $B$ are constants. At $t=5 \mathrm{s}$ the velocity of the car is $31.8 \mathrm{m}{ \mathrm{s}}^{-1}$. Work out the distance travelled in the time interval $0⩽t⩽5$.

A car is moving in a straight line. The acceleration, $a \mathrm{m}{ \mathrm{s}}^{-2}$, at time $t \mathrm{s}$ after the car starts to move is modelled as $a=A(1+4t)$ for $0⩽t<1$ and $a=B\left(30-\frac{10}{t^3}\right)$ for $1⩽t⩽5$, where $A$ and $B$ are constants. By considering the variation of acceleration with time at $t=1$, criticise the model.

A particle $P$ moves on a straight line, starting from rest at a point $O$ of the line. The time after $P$ starts to move is $t \mathrm{s}$, and the particle moves along the line with constant acceleration $\frac{1}{4} \mathrm{m}{ \mathrm{s}}^{-2}$ until it passes through a point $A$ at time $t=8 \mathrm{s}$. After passing through $A$ the velocity of $P$ is $\frac{1}{2}{t}^{\frac{2}{3}} \mathrm{m}{ \mathrm{s}}^{-1}$.

Find the acceleration of $P$ immediately after it passes through $A$. Hence show that the acceleration of $P$ decreases by $\frac{1}{12} \mathrm{m}{ \mathrm{s}}^{-2}$ as it passes through $A$.

A particle $P$ moves on a straight line, starting from rest at a point $O$ of the line. The time after $P$ starts to move is $t \mathrm{s}$, and the particle moves along the line with constant acceleration $\frac{1}{4} \mathrm{m}{ \mathrm{s}}^{-2}$ until it passes through a point $A$ at time $t=8 \mathrm{s}$. After passing through $A$ the velocity of $P$ is $\frac{1}{2}{t}^{\frac{2}{3}} \mathrm{m}{ \mathrm{s}}^{-1}$.

Find the distance moved by $P$ from $t=0 \mathrm{s}$ to $t=27 \mathrm{s}$.

A hockey ball is hit so that it moves in a horizontal straight line with acceleration $a \mathrm{m}{ \mathrm{s}}^{-2}$, along the direction of travel; $a=-0.6t$, where $t$ is the time from when the ball was hit, in seconds. The initial speed of the ball is $14 \mathrm{m}{ \mathrm{s}}^{-1}$. Find the speed of the ball when it has travelled $57.5 \mathrm{m}$.

(Note: you will need an equation solver for this question)

A hockey ball is hit so that it moves in a horizontal straight line with acceleration $a \mathrm{m}{ \mathrm{s}}^{-2}$, along the direction of travel; $a=-0.6t$, where $t$ is the time from when the ball was hit, in seconds. The initial speed of the ball is $14 \mathrm{m}{ \mathrm{s}}^{-1}$. Find the distance that the ball has travelled when the ball is first momentarily stationary.