7. Two ships \(P\) and \(Q\) are travelling at night with constant velocities. At midnight, \(P\) is at the point with position vector \(( 20 \mathbf { i } + 10 \mathbf { j } ) \mathrm { km }\) relative to a fixed origin \(O\). At the same time, \(Q\) is at the point with position vector \(( 14 \mathbf { i } - 6 \mathbf { j } ) \mathrm { km }\). Three hours later, \(P\) is at the point with position vector \(( 29 \mathbf { i } + 34 \mathbf { j } ) \mathrm { km }\). The ship \(Q\) travels with velocity \(12 \mathbf { j } \mathrm {~km} \mathrm {~h} ^ { - 1 }\). At time \(t\) hours after midnight, the position vectors of \(P\) and \(Q\) are \(\mathbf { p } \mathrm { km }\) and \(\mathbf { q } \mathrm { km }\) respectively. Find
- the velocity of \(P\), in terms of \(\mathbf { i }\) and \(\mathbf { j }\),
- expressions for \(\mathbf { p }\) and \(\mathbf { q }\), in terms of \(t\), i and \(\mathbf { j }\).
At time \(t\) hours after midnight, the distance between \(P\) and \(Q\) is \(d \mathrm {~km}\).
- By finding an expression for \(\overrightarrow { P Q }\), show that
$$d ^ { 2 } = 25 t ^ { 2 } - 92 t + 292$$
Weather conditions are such that an observer on \(P\) can only see the lights on \(Q\) when the distance between \(P\) and \(Q\) is 15 km or less. Given that when \(t = 1\), the lights on \(Q\) move into sight of the observer,
- find the time, to the nearest minute, at which the lights on \(Q\) move out of sight of the observer.
Turn over
- In taking off, an aircraft moves on a straight runway \(A B\) of length 1.2 km . The aircraft moves from \(A\) with initial speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It moves with constant acceleration and 20 s later it leaves the runway at \(C\) with speed \(74 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find
- the acceleration of the aircraft,
- the distance \(B C\).
- Two small steel balls \(A\) and \(B\) have mass 0.6 kg and 0.2 kg respectively. They are moving towards each other in opposite directions on a smooth horizontal table when they collide directly. Immediately before the collision, the speed of \(A\) is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(B\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Immediately after the collision, the direction of motion of \(A\) is unchanged and the speed of \(B\) is twice the speed of \(A\). Find
- the speed of \(A\) immediately after the collision,
- the magnitude of the impulse exerted on \(B\) in the collision.
\begin{figure}[h]
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\caption{Figure 1}
\includegraphics[alt={},max width=\textwidth]{94d9432d-1723-4549-ad5e-d4be0f5fd083-018_282_707_278_699}
\end{figure}
A smooth bead \(B\) is threaded on a light inextensible string. The ends of the string are attached to two fixed points \(A\) and \(C\) on the same horizontal level. The bead is held in equilibrium by a horizontal force of magnitude 6 N acting parallel to \(A C\). The bead \(B\) is vertically below \(C\) and \(\angle B A C = \alpha\), as shown in Figure 1. Given that \(\tan \alpha = \frac { 3 } { 4 }\), find - the tension in the string,
- the weight of the bead.
4.
\begin{figure}[h]
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\caption{Figure 2}
\includegraphics[alt={},max width=\textwidth]{94d9432d-1723-4549-ad5e-d4be0f5fd083-019_256_615_280_659}
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A box of mass 2 kg is pulled up a rough plane face by means of a light rope. The plane is inclined at an angle of \(20 ^ { \circ }\) to the horizontal, as shown in Figure 2. The rope is parallel to a line of greatest slope of the plane. The tension in the rope is 18 N . The coefficient of friction between the box and the plane is 0.6 . By modelling the box as a particle, find - the normal reaction of the plane on the box,
- the acceleration of the box.
5. A train is travelling at \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a straight horizontal track. The driver sees a red signal 135 m ahead and immediately applies the brakes. The train immediately decelerates with constant deceleration for 12 s , reducing its speed to \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The driver then releases the brakes and allows the train to travel at a constant speed of \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for a further 15 s . He then applies the brakes again and the train slows down with constant deceleration, coming to rest as it reaches the signal. - Sketch a speed-time graph to show the motion of the train,
- Find the distance travelled by the train from the moment when the brakes are first applied to the moment when its speed first reaches \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
- Find the total time from the moment when the brakes are first applied to the moment when the train comes to rest.
6.
\begin{figure}[h]
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\caption{Figure 3}
\includegraphics[alt={},max width=\textwidth]{94d9432d-1723-4549-ad5e-d4be0f5fd083-022_212_741_287_660}
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A uniform beam \(A B\) has mass 12 kg and length 3 m . The beam rests in equilibrium in a horizontal position, resting on two smooth supports. One support is at the end \(A\), the other at a point \(C\) on the beam, where \(B C = 1 \mathrm {~m}\), as shown in Figure 3. The beam is modelled as a uniform rod. - Find the reaction on the beam at \(C\).
A woman of mass 48 kg stands on the beam at the point \(D\). The beam remains in equilibrium. The reactions on the beam at \(A\) and \(C\) are now equal.
- Find the distance \(A D\).
\includegraphics[max width=\textwidth, alt={}, center]{94d9432d-1723-4549-ad5e-d4be0f5fd083-023_72_58_2632_1873}
7.
\begin{figure}[h]
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\caption{Figure 4}
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\end{figure}
Figure 4 shows a lorry of mass 1600 kg towing a car of mass 900 kg along a straight horizontal road. The two vehicles are joined by a light towbar which is at an angle of \(15 ^ { \circ }\) to the road. The lorry and the car experience constant resistances to motion of magnitude 600 N and 300 N respectively. The lorry's engine produces a constant horizontal force on the lorry of magnitude 1500 N. Find - the acceleration of the lorry and the car,
- the tension in the towbar.
When the speed of the vehicles is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the towbar breaks. Assuming that the resistance to the motion of the car remains of constant magnitude 300 N ,
- find the distance moved by the car from the moment the towbar breaks to the moment when the car comes to rest.
- State whether, when the towbar breaks, the normal reaction of the road on the car is increased, decreased or remains constant. Give a reason for your answer.