The diagram shows a vertical cross-section, $ABCD$, of a fixed surface. $AB$ and $CD$ are smooth curves and $BC$ is a rough horizontal surface. $A$ is at a vertical height $3.2 \mathrm{m}$ above $BC$. A particle of mass $2 \mathrm{kg}$ is released from $A$ and travels along the surface to $D$.

Find the speed of the particle at $B$.

The diagram shows a vertical cross-section, $ABCD$, of a fixed surface. $AB$ and $CD$ are smooth curves and $BC$ is a rough horizontal surface. $A$ is at a vertical height $3.2 \mathrm{m}$ above $BC$. A particle of mass $2 \mathrm{kg}$ is released from $A$ and travels along the surface to $D$.

Given that the particle reaches $C$ with a speed of $4 \mathrm{m}{ \mathrm{s}}^{-1}$, find the work done against the resistance to motion as the particle moves from $B$ to $C$.

The diagram shows a vertical cross-section, $ABCD$, of a fixed surface. $AB$ and $CD$ are smooth curves and $BC$ is a rough horizontal surface. $A$ is at a vertical height $3.2 \mathrm{m}$ above $BC$. A particle of mass $2 \mathrm{kg}$ is released from $A$ and travels along the surface to $D$.

The particle reaches the point $D$. Find the maximum vertical height of $D$ above $BC$.

A car of mass $2000 \mathrm{kg}$ climbs a straight hill, $ABC$, which makes an angle $\theta$ with the horizontal, where $\sin \theta =\frac{1}{16}$. For the motion from $A$ to $B$, the work done by the car's engine is $256 \mathrm{kJ}$ and the resistance to motion is $R \mathrm{N}$. The length of $AB$ is $200 \mathrm{m}$. The speed of the car is $20 \mathrm{m}{ \mathrm{s}}^{-1}$ at $A$ and $16 \mathrm{m}{ \mathrm{s}}^{-1}$ at $B$.

Find the value of $R$.

A car of mass $2000 \mathrm{kg}$ climbs a straight hill, $ABC$, which makes an angle $\theta$ with the horizontal, where $\sin \theta =\frac{1}{16}$. For the motion from $A$ to $B$, the work done by the car's engine is $256 \mathrm{kJ}$ and the resistance to motion is $R \mathrm{N}$. The length of $AB$ is $200 \mathrm{m}$. The speed of the car is $20 \mathrm{m}{ \mathrm{s}}^{-1}$ at $A$ and $16 \mathrm{m}{ \mathrm{s}}^{-1}$ at $B$.

From $B$ to $C$, the work done by the engine is $388 \mathrm{kJ}$. The resistance to motion remains the same as that between $A$ and $B$. The speed of the car at $C$ is $12 \mathrm{m}{ \mathrm{s}}^{-1}$. Find the distance $BC$.

A particle, $P$, of mass $4 \mathrm{kg}$ is projected with speed $9 \mathrm{m}{ \mathrm{s}}^{-1}$ along rough horizontal ground. The coefficient of friction between $P$ and the ground is $0.4$. After $7 \mathrm{m}$ it strikes a stationary particle, $Q$, of mass $3 \mathrm{kg}$. The coefficient of friction between $Q$ and the ground is also $0.4$. $Q$ comes to rest after $2 \mathrm{m}$.

Show that the speed of $P$ immediately before the collision is $5 \mathrm{m}{ \mathrm{s}}^{-1}$ and find the speed of $Q$ immediately after the collision.

A particle, $P$, of mass $4 \mathrm{kg}$ is projected with speed $9 \mathrm{m}{ \mathrm{s}}^{-1}$ along rough horizontal ground. The coefficient of friction between $P$ and the ground is $0.4$. After $7 \mathrm{m}$ it strikes a stationary particle, $Q$, of mass $3 \mathrm{kg}$. The coefficient of friction between $Q$ and the ground is also $0.4$. $Q$ comes to rest after $2 \mathrm{m}$.

Find the distance between $Q$ and $P$ when both have come to rest.

A lorry of mass $24000 \mathrm{kg}$ is travelling up a hill which is inclined at $3°$ to the horizontal. The power developed by the lorry's engine is constant, and there is a constant resistance to motion of $3200 \mathrm{N}$.

When the speed of the lorry is $25 \mathrm{m}{ \mathrm{s}}^{-1}$, its acceleration is $0.2 \mathrm{m}{ \mathrm{s}}^{-2}$. Find the power developed by the lorry's engine.

A lorry of mass $24000 \mathrm{kg}$ is travelling up a hill which is inclined at $3°$ to the horizontal. The power developed by the lorry's engine is constant, and there is a constant resistance to motion of $3200 \mathrm{N}$.

Find the steady speed at which the lorry moves up the hill if the power is $500 \mathrm{kW}$ and the resistance remains $3200 \mathrm{N}$.