OCR MEI M1 (Mechanics 1) 2013 January

1 Fig. 1 shows a block of mass 3 kg on a plane which is inclined at an angle of \(30 ^ { \circ }\) to the horizontal.
A force \(P \mathrm {~N}\) is applied to the block parallel to the plane in the upwards direction.
The plane is rough so that a frictional force of 10 N opposes the motion.
The block is moving at constant speed up the plane. \begin{figure}[h] \end{figure}

1. Mark and label all the forces acting on the block.
2. Calculate the magnitude of the normal reaction of the plane on the block.
3. Calculate the magnitude of the force \(P\).

2 In this question, the unit vectors \(\binom { 1 } { 0 }\) and \(\binom { 0 } { 1 }\) are in the directions east and north.
Distance is measured in metres and time, \(t\), in seconds.
A radio-controlled toy car moves on a flat horizontal surface. A child is standing at the origin and controlling the car.
When \(t = 0\), the displacement of the car from the origin is \(\binom { 0 } { - 2 } \mathrm {~m}\), and the car has velocity \(\binom { 2 } { 0 } \mathrm {~ms} ^ { - 1 }\). The acceleration of the car is constant and is \(\binom { - 1 } { 1 } \mathrm {~ms} ^ { - 2 }\).

1. Find the velocity of the car at time \(t\) and its speed when \(t = 8\).
2. Find the distance of the car from the child when \(t = 8\).

3 Fig. 3 shows two people, Sam and Tom, pushing a car of mass 1000 kg along a straight line \(l\) on level ground. Sam pushes with a constant horizontal force of 300 N at an angle of \(30 ^ { \circ }\) to the line \(l\).
Tom pushes with a constant horizontal force of 175 N at an angle of \(15 ^ { \circ }\) to the line \(l\). \begin{figure}[h] \end{figure}

1. The car starts at rest and moves with constant acceleration. After 6 seconds it has travelled 7.2 m . Find its acceleration.
2. Find the resistance force acting on the car along the line \(l\).
3. The resultant of the forces exerted by Sam and Tom is not in the direction of the car's acceleration. Explain briefly why.

4 A particle is travelling along a straight line with constant acceleration. \(\mathrm { P } , \mathrm { O }\) and Q are points on the line, as illustrated in Fig. 4. The distance from P to O is 5 m and the distance from O to Q is 30 m . \begin{figure}[h] \end{figure} Initially the particle is at O . After 10 s , it is at Q and its velocity is \(9 \mathrm {~ms} ^ { - 1 }\) in the direction \(\overrightarrow { \mathrm { OQ } }\).

1. Find the initial velocity and the acceleration of the particle.
2. Prove that the particle is never at P .

5 Ali is throwing flat stones onto water, hoping that they will bounce, as illustrated in Fig. 5.
Ali throws one stone from a height of 1.225 m above the water with initial speed \(20 \mathrm {~ms} ^ { - 1 }\) in a horizontal direction. Air resistance should be neglected. \begin{figure}[h] \end{figure}

1. Find the time it takes for the stone to reach the water.
2. Find the speed of the stone when it reaches the water and the angle its trajectory makes with the horizontal at this time.

6 The speed of a 100 metre runner in \(\mathrm { ms } ^ { - 1 }\) is measured electronically every 4 seconds.
The measurements are plotted as points on the speed-time graph in Fig. 6. The vertical dotted line is drawn through the runner's finishing time. Fig. 6 also illustrates Model P in which the points are joined by straight lines. \begin{figure}[h] \end{figure}

1. Use Model P to estimate
   (A) the distance the runner has gone at the end of 12 seconds,
   (B) how long the runner took to complete 100 m . A mathematician proposes Model Q in which the runner's speed, \(v \mathrm {~ms} ^ { - 1 }\) at time \(t \mathrm {~s}\), is given by $$v = \frac { 5 } { 2 } t - \frac { 1 } { 8 } t ^ { 2 }$$
2. Verify that Model Q gives the correct speed for \(t = 8\).
3. Use Model Q to estimate the distance the runner has gone at the end of 12 seconds.
4. The runner was timed at 11.35 seconds for the 100 m . Which model places the runner closer to the finishing line at this time?
5. Find the greatest acceleration of the runner according to each model.