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.

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 value of $t$ when the ball has travelled $40 \mathrm{m}$.

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

Two particles $A$ and $B$ start to move at the same instant from a point $O$. The particles move in the same direction along the same straight line. The acceleration of $A$ at time $t \mathrm{s}$ after starting to move is $a \mathrm{m}{ \mathrm{s}}^{-2}$, where $a=0.05-0.0002t$.  Find $A\text{'}s$ velocity when $t=200 \mathrm{s}$ and when $t=500 \mathrm{s}$.

Two particles $A$ and $B$ start to move at the same instant from a point $O$. The particles move in the same direction along the same straight line. The acceleration of $A$ at time $t \mathrm{s}$ after starting to move is $a \mathrm{m}{ \mathrm{s}}^{-2}$, where $a=0.05-0.0002t$. $B$ moves with constant acceleration for the first $200 \mathrm{s}$ and has the same velocity as $A$ when $t=200 \mathrm{s}$. $B$ moves with constant retardation from $t=200 \mathrm{s}$ to $t=500 \mathrm{s}$ and has the same velocity as $A$ when $t=500 \mathrm{s}$. Find the distance between $A$ and $B$ when $t=500 \mathrm{s}$.

A vehicle is moving in a straight line. The velocity $v \mathrm{m}{ \mathrm{s}}^{-1}$ at time $t \mathrm{s}$ after the vehicle starts is given by $v=A\left(t-0.05t^2\right)$ for $0⩽t⩽15$ and $v=\frac{B}{t^2}$ for $t⩾15$, where $A$ and $B$ are constants. The distance travelled by the vehicle between $t=0 \mathrm{s}$ and $t=15 \mathrm{s}$ is $225 \mathrm{m}$.

Find the value of $A$ and show that $B=3375$.

A vehicle is moving in a straight line. The velocity $v \mathrm{m}{ \mathrm{s}}^{-1}$ at time $t \mathrm{s}$ after the vehicle starts is given by $v=A\left(t-0.05t^2\right)$ for $0 \mathrm{s}⩽t⩽15 \mathrm{s}$ and $v=\frac{B}{t^2}$ for $t⩾15$, where $A$ and $B$ are constants. The distance travelled by the vehicle between $t=0 \mathrm{s}$ and $t=15 \mathrm{s}$ is $225 \mathrm{m}$. Find an expression in terms of $t$ for the total distance travelled by the vehicle when $t⩾15 \mathrm{s}$.

A vehicle is moving in a straight line. The velocity $v \mathrm{m}{ \mathrm{s}}^{-1}$ at time $t \mathrm{s}$ after the vehicle starts is given by $v=A\left(t-0.05t^2\right)$ for $0 \mathrm{s}⩽t⩽15 \mathrm{s}$ and $v=\frac{B}{t^2}$ for $t⩾15 \mathrm{s}$, where $A$ and $B$ are constants. The distance travelled by the vehicle between $t=0 \mathrm{s}$ and $t=15 \mathrm{s}$ is $225 \mathrm{m}$.

Find the speed of the vehicle when it has travelled a total distance of $315 \mathrm{m}$.