A small van of mass $1600 \mathrm{kg}$accelerates from rest in a straight line along a horizontal road. The resistance from friction and air resistance is $2400 \mathrm{N}$ throughout the motion. The engine works at a constant rate of $48 \mathrm{kW}$.

Write down an expression for the acceleration of the van when it is travelling at $v{ \mathrm{ms}}^{-1}$.

A small van of mass $1600 \mathrm{kg}$accelerates from rest in a straight line along a horizontal road. The resistance from friction and air resistance is $2400 \mathrm{N}$ throughout the motion. The engine works at a constant rate of $48 \mathrm{kW}$.

Explain why the power cannot be constant.

A powerboat of mass $500 \mathrm{kg}$ travels in a straight line at its maximum velocity against a resistance of $15 \mathrm{N}$. The engine of the powerboat has a maximum power output of $750 \mathrm{W}$.

Find the maximum velocity.

A powerboat of mass $500 \mathrm{kg}$ travels in a straight line at its maximum velocity against a resistance of $15 \mathrm{N}$. The engine of the powerboat has a maximum power output of $750 \mathrm{W}$.

In different weather conditions the same powerboat has a maximum velocity of only 25 $m{s}^{-1}$

State what has changed in the model and give the new value of this quantity.

Car $A$, of mass $1250 \mathrm{kg}$, is travelling along a straight horizontal road at speed $10 \mathrm{m}{ \mathrm{s}}^{-1}$. The engine works at a constant rate of $25 \mathrm{kW}$ and the resistance is a constant $500 \mathrm{N}$. After $5 \mathrm{s}$ the speed of the car has increased to $v \mathrm{m}{ \mathrm{s}}^{-1}$.

Use the work-energy principle to find the amount of energy that is dissipated and, hence, find the distance travelled in the $5 \mathrm{s}$.

Car $A$, of mass $1250 \mathrm{kg}$, is travelling along a straight horizontal road at speed $10 \mathrm{m}{ \mathrm{s}}^{-1}$. The engine works at a constant rate of $25 \mathrm{kW}$ and the resistance is a constant $500 \mathrm{N}$. After $5 \mathrm{s}$ the speed of the car has increased to $v \mathrm{m}{ \mathrm{s}}^{-1}$.

Find an expression for the acceleration at time $5 \mathrm{s}$ as a function of $v$ and show that the acceleration is not constant.

Car $A$, of mass $1250 \mathrm{kg}$, is travelling along a straight horizontal road at speed $10 \mathrm{m}{ \mathrm{s}}^{-1}$. The engine works at a constant rate of $25 \mathrm{kW}$ and the resistance is a constant $500 \mathrm{N}$. After $5 \mathrm{s}$ the speed of the car has increased to $v \mathrm{m}{ \mathrm{s}}^{-1}$.

Car B travels along the same road, starting with speed $10 m{s}^{-1}$ and accelerating at a constant rate for$5s$ . After$5s$  the two cars have the same speed and also have the same acceleration as one another.

Show that $v$ must satisfy the equation $v^2-8v=100$ and, hence, find the speed of the cars at the end of the $5 \mathrm{s}$.