OCR MEI M1 (Mechanics 1)

1 Fig. 4 illustrates a straight horizontal road. \(A\) and \(B\) are points on the road which are 215 metres apart and \(M\) is the mid-point of AB . When a car passes A its speed is \(12 \mathrm {~ms} ^ { - 1 }\) in the direction AB . It then accelerates uniformly and when it reaches \(B\) its speed is \(31 \mathrm {~ms} ^ { - 1 }\). \begin{figure}[h] \end{figure}

1. Find the car's acceleration.
2. Find how long it takes the car to travel from A to B .
3. Find how long it takes the car to travel from A to M .
4. Explain briefly, in terms of the speed of the car, why the time taken to travel from A to M is more than half the time taken to travel from A to B .

2 In this question, air resistance should be neglected.
Fig. 2 illustrates the flight of a golf ball. The golf ball is initially on the ground, which is horizontal. \begin{figure}[h] \end{figure} It is hit and given an initial velocity with components of \(15 \mathrm {~ms} ^ { - 1 }\) in the horizontal direction and \(20 \mathrm {~ms} ^ { - 1 }\) in the vertical direction.

1. Find its initial speed.
2. Find the ball's flight time and range, \(R \mathrm {~m}\).
3. (A) Show that the range is the same if the components of the initial velocity of the ball are \(20 \mathrm {~ms} ^ { - 1 }\) in the horizontal direction and \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the vertical direction.
   (B) State, justifying your answer, whether the range is the same whenever the ball is hit with the same initial speed.

3 A particle is moving along a straight line and its position is relative to an origin on the line. At time \(t \mathrm {~s}\), the particle's acceleration, \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), is given by $$a = 6 t - 12 .$$ At \(t = 0\) the velocity of the particle is \(+ 9 \mathrm {~ms} ^ { - 1 }\) and its position is - 2 m .

1. Find an expression for the velocity of the particle at time \(t \mathrm {~s}\) and verify that it is stationary when \(t = 3\).
2. Find the position of the particle when \(t = 2\).

4 Fig. 4 illustrates points \(\mathrm { A } , \mathrm { B }\) and C on a straight race track. The distance AB is 300 m and AC is 500 m .
A car is travelling along the track with uniform acceleration. \begin{figure}[h] \end{figure} Initially the car is at A and travelling in the direction AB with speed \(5 \mathrm {~ms} ^ { - 1 }\). After 20s it is at C .

1. Find the acceleration of the car.
2. Find the speed of the car at B and how long it takes to travel from A to B .

5 A particle is moving along a straight line and its position is relative to an origin on the line. At time \(t \mathrm {~s}\), the particle's acceleration, \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), is given by $$a = 6 t - 12 .$$ At \(t = 0\) the velocity of the particle is \(+ 9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its position is - 2 m .

1. Find an expression for the velocity of the particle at time \(t \mathrm {~s}\) and verify that it is stationary when \(t = 3\).
2. Find the position of the particle when \(t = 2\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{b9e41fac-9f4b-4165-af03-67ebdcb326de-3_349_987_375_623} \captionsetup{labelformat=empty} \caption{Fig. 4}
   \end{figure} Particles P and Q move in the same straight line. Particle P starts from rest and has a constant acceleration towards \(Q\) of \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Particle \(Q\) starts 125 m from \(P\) at the same time and has a constant speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) away from \(P\). The initial values are shown in Fig. 4.
3. Write down expressions for the distances travelled by P and by Q at time \(t\) seconds after the start of the motion.
4. How much time does it take for P to catch up with Q and how far does P travel in this time?