Edexcel FM1 AS (Further Mechanics 1 AS) Specimen

1. A small ball of mass 0.1 kg is dropped from a point which is 2.4 m above a horizontal floor. The ball falls freely under gravity, strikes the floor and bounces to a height of 0.6 m above the floor. The ball is modelled as a particle.
   1. Show that the coefficient of restitution between the ball and the floor is 0.5
   2. Find the height reached by the ball above the floor after it bounces on the floor for the second time.
   3. By considering your answer to (b), describe the subsequent motion of the ball.

1. A small stone of mass 0.5 kg is thrown vertically upwards from a point A with an initial speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The stone first comes to instantaneous rest at the point B which is 20 m vertically above the point A . As the stone moves it is subject to air resistance. The stone is modelled as a particle.
   1. Find the energy lost due to air resistance by the stone, as it moves from A to B
   The air resistance is modelled as a constant force of magnitude \(R\) newtons.
2. Find the value of R .
3. State how the model for air resistance could be refined to make it more realistic.

1. \hspace{0pt} [In this question use \(\mathrm { g } = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) ]

A jogger of mass 60 kg runs along a straight horizontal road at a constant speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The total resistance to the motion of the jogger is modelled as a constant force of magnitude 30 N .

1. Find the rate at which the jogger is working. The jogger now comes to a hill which is inclined to the horizontal at an angle \(\alpha\), where \(\sin \alpha = \frac { 1 } { 15 }\). Because of the hill, the jogger reduces her speed to \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and maintains this constant speed as she runs up the hill. The total resistance to the motion of the jogger from non-gravitational forces continues to be modelled as a constant force of magnitude 30 N .
2. Find the rate at which she has to work in order to run up the hill at \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. A particle P of mass 3 m is moving in a straight line on a smooth horizontal table. A particle \(Q\) of mass \(m\) is moving in the opposite direction to \(P\) along the same straight line. The particles collide directly. Immediately before the collision the speed of P is u and the speed of Q is 2 u . The velocities of P and Q immediately after the collision, measured in the direction of motion of P before the collision, are V and W respectively. The coefficient of restitution between P and Q is e .
   1. Find an expression for v in terms of u and e .
   Given that the direction of motion of P is changed by the collision,
2. find the range of possible values of e.
3. Show that \(\mathrm { w } = \frac { \mathrm { u } } { 4 } ( 1 + 9 \mathrm { e } )\). Following the collision with P , the particle Q then collides with and rebounds from a fixed vertical wall which is perpendicular to the direction of motion of \(Q\). The coefficient of restitution between \(Q\) and the wall is \(f\).
   Given that \(\mathrm { e } = \frac { 5 } { 9 }\), and that P and Q collide again in the subsequent motion,
4. find the range of possible values of f .

   VIIIV SIHI NI JIIIM ION OC

   VIIIV SIHI NI JIHM I I ON OC

   VIAV SIHI NI JIIIM I ON OC

   \section*{Q uestion 4 continued}