Jan Dangerfield, Stuart Haring and, Julian Gilbey Solutions for Exercise 4: EXERCISE 7B

Mariam is moving on a sledge at $2 \mathrm{m}{ \mathrm{s}}^{-1}$. The combined mass of Mariam and the sledge is $40 \mathrm{kg}$. Sarah, who has mass $60 \mathrm{kg}$, runs up behind the sledge and jumps onto it. The sledge continues in the same straight line with speed $2.3 \mathrm{m}{ \mathrm{s}}^{-1}$.

What assumption have you made regarding Sarah's velocity?

A simplified model of the launch of a space shuttle is as follows. The shuttle is attached to two rocket boosters each of which contains a fuel tank. The launch is vertical and in a straight line. The initial total mass is $1 \mathrm{million} \mathrm{kg}$. The mass of the shuttle is $60000 \mathrm{kg}$, the mass of each rocket booster is $20000 \mathrm{kg}$ and the mass of the fuel in each rocket booster is $450000 \mathrm{kg}$. The rocket boosters accelerate the shuttle (and themselves) to a speed of $1500 \mathrm{m}{ \mathrm{s}}^{-1}$. At this time all the fuel in the rocket boosters has been used up and the rocket boosters are detached. The rocket boosters have speed $0 \mathrm{m}{ \mathrm{s}}^{-1}$ immediately after they are detached.

Use conservation of momentum to show that the speed of the shuttle immediately after the rocket boosters are detached is $2500 \mathrm{m}{ \mathrm{s}}^{-1}$.

A simplified model of the launch of a space shuttle is as follows. The shuttle is attached to two rocket boosters each of which contains a fuel tank. The launch is vertical and in a straight line. The initial total mass is $1 \mathrm{million} \mathrm{kg}$. The mass of the shuttle is $60000 \mathrm{kg}$, the mass of each rocket booster is $20000 \mathrm{kg}$ and the mass of the fuel in each rocket booster is $450000 \mathrm{kg}$. The rocket boosters accelerate the shuttle (and themselves) to a speed of $1500 \mathrm{m}{ \mathrm{s}}^{-1}$. At this time all the fuel in the rocket boosters has been used up and the rocket boosters are detached. The rocket boosters have speed $0 \mathrm{m}{ \mathrm{s}}^{-1}$ immediately after they are detached.

Suppose that instead just the first rocket booster is used initially to accelerate the shuttle (with both rocket boosters) to a speed of $500 \mathrm{m}{ \mathrm{s}}^{-1}$. At this time all the fuel in the first rocket booster has been used up and it is detached (with speed $0 \mathrm{m}{ \mathrm{s}}^{-1}$). The second rocket booster is still full of fuel.

Show that the speed of the shuttle (with the remaining rocket booster) is $518.9 \mathrm{m}{ \mathrm{s}}^{-1}$ immediately after the first rocket booster is detached.

A simplified model of the launch of a space shuttle is as follows. The shuttle is attached to two rocket boosters each of which contains a fuel tank. The launch is vertical and in a straight line. The initial total mass is $1 \mathrm{million} \mathrm{kg}$. The mass of the shuttle is $60000 \mathrm{kg}$, the mass of each rocket booster is $20000 \mathrm{kg}$ and the mass of the fuel in each rocket booster is $450000 \mathrm{kg}$. The rocket boosters accelerate the shuttle (and themselves) to a speed of $1500 \mathrm{m}{ \mathrm{s}}^{-1}$. At this time all the fuel in the rocket boosters has been used up and the rocket boosters are detached. The rocket boosters have speed $0 \mathrm{m}{ \mathrm{s}}^{-1}$ immediately after they are detached.

Suppose that instead just the first rocket booster is used initially to accelerate the shuttle (with both rocket boosters) to a speed of $500 \mathrm{m}{ \mathrm{s}}^{-1}$. At this time all the fuel in the first rocket booster has been used up and it is detached (with speed $0 \mathrm{m}{ \mathrm{s}}^{-1}$). The second rocket booster is still full of fuel.

The second rocket booster is then used to accelerate the shuttle (and itself). When all the fuel in the second rocket booster has been used up it is detached (with speed $0{ \mathrm{ms}}^{-1}$). After the second rocket booster has been detached, the speed of the shuttle is $2500 \mathrm{m}{ \mathrm{s}}^{-1}$.

Find the speed of the shuttle (and rocket booster) just before the second rocket booster was detached.

A car is towing a caravan at $v \mathrm{m}{ \mathrm{s}}^{-1}$ in a straight line along a horizontal road. The mass of the car is $m \mathrm{kg}$ and the mass of the caravan is $\mathrm{km} \mathrm{kg}$. The caravan becomes detached from the car. Immediately after the separation the car has speed $\alpha v \mathrm{m}{ \mathrm{s}}^{-1}$ and the caravan has speed $0.5v \mathrm{m}{ \mathrm{s}}^{-1}$ in the same direction as the car.

Show that if $\alpha =1.3$ then $k=0.6$.

A car is towing a caravan at $v \mathrm{m}{ \mathrm{s}}^{-1}$ in a straight line along a horizontal road. The mass of the car is $m \mathrm{kg}$ and the mass of the caravan is $\mathrm{km} \mathrm{kg}$. The caravan becomes detached from the car. Immediately after the separation the car has speed $\alpha v \mathrm{m}{ \mathrm{s}}^{-1}$ and the caravan has speed $0.5v \mathrm{m}{ \mathrm{s}}^{-1}$ in the same direction as the car.

Find an expression for $k$ in terms of a general value of $\alpha$.

A particle of mass $0.3 \mathrm{kg}$ is travelling at speed $2 \mathrm{m}{ \mathrm{s}}^{-1}$ when it collides with a particle of mass $0.5 \mathrm{kg}$ travelling at speed $1 \mathrm{m}{ \mathrm{s}}^{-1}$. After the impact the first particle has speed $v \mathrm{m}{ \mathrm{s}}^{-1}$ and the second particle has speed $\left(\mathrm{v}+1\right){\mathrm{ms}}^{-1}$

By considering the directions in which the particles could be moving before and after the impact, find the possible values for the speed of the first particle after the impact.

A particle of mass $0.3 \mathrm{kg}$ Is travelling at speed $2 \mathrm{m}{ \mathrm{s}}^{-1}$ When it collides with a particle of mass $0.5 \mathrm{kg}$ Travelling at speed $1 \mathrm{m}{ \mathrm{s}}^{-1}$. After the impact the first particle has speed $v \mathrm{m}{ \mathrm{s}}^{-1}$ And the second particle has speed $\left(\mathrm{v}+1\right){\mathrm{ms}}^{-1}$

You are given that $v$ Is the smallest of these possible speeds.

State whether the particles were travelling in the same direction or in opposite directions before the impact.