OCR M1 (Mechanics 1) 2012 January

1 Particles \(P\) and \(Q\), of masses 0.3 kg and 0.5 kg respectively, are moving in the same direction along the same straight line on a smooth horizontal surface. \(P\) is moving with speed \(2.2 \mathrm {~ms} ^ { - 1 }\) and \(Q\) is moving with speed \(0.8 \mathrm {~ms} ^ { - 1 }\) immediately before they collide. In the collision, the speed of \(P\) is reduced by \(50 \%\) and its direction of motion is unchanged.

1. Calculate the speed of \(Q\) immediately after the collision.
2. Find the distance \(P Q\) at the instant 3 seconds after the collision.

2 In the sport of curling, a heavy stone is projected across a horizontal ice surface. One player projects a stone of weight 180 N , which moves 36 m in a straight line and comes to rest 24 s after the instant of projection. The only horizontal force acting on the stone after its projection is a constant frictional force between the stone and the ice.

1. Calculate the deceleration of the stone.
2. Find the magnitude of the frictional force acting on the stone, and calculate the coefficient of friction between the stone and the ice.

3 A car is travelling along a straight horizontal road with velocity \(32.5 \mathrm {~ms} ^ { - 1 }\). The driver applies the brakes and the car decelerates at \(( 8 - 0.6 t ) \mathrm { ms } ^ { - 2 }\), where \(t \mathrm {~s}\) is the time which has elapsed since the brakes were first applied.

1. Show that, while the car is decelerating, its velocity is \(\left( 32.5 - 8 t + 0.3 t ^ { 2 } \right) \mathrm { m } \mathrm { s } ^ { - 1 }\).
2. Find the time taken to bring the car to rest.
3. Show that the distance travelled while the car is decelerating is 75 m .

4
\includegraphics[max width=\textwidth, alt={}, center]{2b3457b6-1fe9-4e67-91d4-a8bc4a5b1709-2_325_481_1699_792} Three horizontal forces of magnitudes \(8 \mathrm {~N} , 15 \mathrm {~N}\) and 20 N act at a point. The 8 N and 15 N forces are at right angles. The 20 N force makes an angle of \(150 ^ { \circ }\) with the 8 N force and an angle of \(120 ^ { \circ }\) with the 15 N force (see diagram).

1. Calculate the components of the resultant force in the directions of the 8 N and 15 N forces.
2. Calculate the magnitude of the resultant force, and the angle it makes with the direction of the 8 N force. The directions in which the three horizontal forces act can be altered.
3. State the greatest and least possible magnitudes of the resultant force.

5
\includegraphics[max width=\textwidth, alt={}, center]{2b3457b6-1fe9-4e67-91d4-a8bc4a5b1709-3_394_789_251_639} The diagram shows the ( \(t , v\) ) graph of an athlete running in a straight line on a horizontal track in a 100 m race. He starts from rest and has constant acceleration until he reaches a speed of \(15 \mathrm {~ms} ^ { - 1 }\) when \(t = T\). He maintains this constant speed until he decelerates at a constant rate of \(1.75 \mathrm {~ms} ^ { - 2 }\) for the final 4 s of the race. He completes the race in 10 s .

1. Calculate \(T\). The athlete races against a robot which has a displacement from the starting line of \(\left( 3 t ^ { 2 } - 0.2 t ^ { 3 } \right) \mathrm { m }\), at time \(t \mathrm {~s}\) after the start of the race.
2. Show that the speed of the robot is \(15 \mathrm {~ms} ^ { - 1 }\) when \(t = 5\).
3. Find the value of \(t\) for which the decelerations of the robot and the athlete are equal.
4. Verify that the athlete and the robot reach the finish line simultaneously.

6 A particle \(P\) of mass 0.3 kg is projected upwards along a line of greatest slope from the foot of a plane inclined at \(30 ^ { \circ }\) to the horizontal. The initial speed of \(P\) is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the coefficient of friction is 0.15 . The particle \(P\) comes to instantaneous rest before it reaches the top of the plane.

1. Calculate the distance \(P\) moves up the plane.
2. Find the time taken by \(P\) to return from its highest position on the plane to the foot of the plane.
3. Calculate the change in the momentum of \(P\) between the instant that \(P\) leaves the foot of the plane and the instant that \(P\) returns to the foot of the plane.

7
\includegraphics[max width=\textwidth, alt={}, center]{2b3457b6-1fe9-4e67-91d4-a8bc4a5b1709-4_369_508_246_781} Particles \(P\) and \(Q\), of masses \(m \mathrm {~kg}\) and 0.05 kg respectively, are attached to the ends of a light inextensible string which passes over a smooth pulley. \(Q\) is attached to a particle \(R\) of mass 0.45 kg by a light inextensible string. The strings are taut, and the portions of the strings not in contact with the pulley are vertical. \(P\) is in contact with a horizontal surface when the particles are released from rest (see diagram). The tension in the string \(Q R\) is 2.52 N during the descent of \(R\).

1. (a) Find the acceleration of \(R\) during its descent.
   (b) By considering the motion of \(Q\), calculate the tension in the string \(P Q\) during the descent of \(R\).
2. Find the value of \(m\).
   \(R\) strikes the surface 0.5 s after release and does not rebound. During their subsequent motion, \(P\) does not reach the pulley and \(Q\) does not reach the surface.
3. Calculate the greatest height of \(P\) above the surface.