1. 1
   A stone is thrown horizontally with a speed of 15 m s⁻¹ from the top of a vertical cliff 45 m above the sea. Air resistance is negligible; take g = 9.81 m s⁻².

   (a)

   How long does the stone take to reach the sea?

   [1 mark]

   • A3.0 s
   • B2.1 s
   • C9.2 s
   • D4.6 s

   (b)

   What is the magnitude of the stone's velocity as it hits the sea?

   [1 mark]

   • A29.7 m s⁻¹
   • B33.3 m s⁻¹
   • C15.0 m s⁻¹
   • D44.7 m s⁻¹

   (c)

   Calculate the horizontal distance from the base of the cliff at which the stone lands.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A firework shell of mass 0.80 kg, initially at rest in mid-air, explodes into two fragments. Fragment P has mass 0.30 kg and moves off at 40 m s⁻¹. Fragment Q has the remaining mass and moves off in the opposite direction.

   (a)

   What is the speed of fragment Q immediately after the explosion?

   [1 mark]

   • A40 m s⁻¹
   • B66.7 m s⁻¹
   • C24 m s⁻¹
   • D15 m s⁻¹

   (b)

   Which statement about this explosion is correct?

   [1 mark]

   • AMomentum is conserved and kinetic energy is conserved
   • BMomentum is not conserved but kinetic energy is conserved
   • CNeither momentum nor kinetic energy is conserved
   • DMomentum is conserved and the total kinetic energy increases

   (c)

   Calculate the total kinetic energy of the two fragments immediately after the explosion.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A cyclist and bicycle of total mass 78 kg free-wheel from rest down a straight hill inclined at 5.0° to the horizontal. Over a distance of 200 m measured along the slope, resistive forces (air resistance and rolling friction) do a total of 3800 J of negative work on the cyclist.

   (a)

   Show that the loss in gravitational potential energy of the cyclist over this 200 m is about 1.3 × 10⁴ J, and hence calculate the cyclist's speed at the bottom of the hill.

   [3 marks]

   (b)

   At the bottom of the hill the cyclist pedals along a flat road, maintaining the speed found in (a) against a constant total resistive force of 25 N. (i) Calculate the cyclist's power output. (ii) The cyclist's legs supply energy at a rate of 520 W. Calculate the efficiency of this transfer of energy into useful power output.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A solid uniform disc of mass 2.0 kg and radius 0.15 m rotates about a fixed, frictionless horizontal axis through its centre, perpendicular to its plane. Its moment of inertia about this axis is I = ½MR². A string wrapped around its rim is pulled with a constant force of 6.0 N, tangential to the rim, starting from rest.

   (a)

   Show that the moment of inertia of the disc is 0.0225 kg m², calculate the angular acceleration produced by the string, and determine the angular velocity of the disc after it has turned through 5.0 complete revolutions from rest.

   [6 marks]

   (b)

   Using the work–energy theorem, show that the rotational kinetic energy gained by the disc is consistent with your answer to (a), find the linear speed of a point on the rim of the disc at this instant, and explain what this consistency illustrates about energy conservation for the rotating disc.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A spacecraft travels at a constant speed of 0.80c relative to Earth between two space stations that are at rest relative to Earth. Observers on Earth measure the journey to take 15.0 years.

   (a)

   What is the value of the Lorentz factor γ for the spacecraft?

   [1 mark]

   • A1.67
   • B0.60
   • C1.25
   • D2.00

   (b)

   Which time interval is the proper time for this journey?

   [1 mark]

   • AThe 15.0 years measured by clocks on Earth, because Earth is taken to be at rest
   • BThe 9.0 years shown on a clock carried on the spacecraft, since that clock is present at both the departure and arrival events
   • CBoth the 15.0 years and the 9.0 years are proper times
   • DNeither interval is a proper time, since both frames are inertial

   (c)

   Calculate the distance between the two space stations as measured by an observer on the spacecraft.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A small ball of mass 0.40 kg is attached to a light, inextensible string of length 0.60 m and swung so that it moves at constant speed in a horizontal circle, with the string making a constant angle of 25° with the vertical (a conical pendulum). Take g = 9.81 m s⁻².

   (a)

   What is the tension in the string?

   [1 mark]

   • A3.9 N
   • B9.3 N
   • C4.3 N
   • D1.7 N

   (b)

   Which force (or component of a force) provides the centripetal force on the ball?

   [1 mark]

   • AThe weight of the ball
   • BThe full tension in the string
   • CThe vertical component of the tension
   • DThe horizontal component of the tension

   (c)

   Calculate the speed of the ball as it moves around the circle.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A student of moment of inertia 3.6 kg m² about a vertical axis sits at rest on a frictionless rotating stool, holding a 1.5 kg mass in each outstretched hand, 0.80 m from the axis. A friend spins the student so that the system rotates at 2.0 rad s⁻¹. The student then pulls both masses in to a new distance of 0.25 m from the axis, without any external torque acting on the system.

   (a)

   Calculate the moment of inertia of the student-and-masses system about the axis before the masses are pulled in, and the angular momentum of the system at that time.

   [3 marks]

   (b)

   Calculate the new angular speed of the system after the masses are pulled in, and the increase in the total rotational kinetic energy of the system. State the source of this extra energy.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A space probe is launched from Earth. During its first propulsion stage, its engines provide a uniform acceleration of 25 m s⁻² for 60 s, starting from rest, before cutting off. Much later in its journey, after further propulsion by a separate engine system, the probe travels at a constant speed of 0.95c relative to Earth for a final leg of the journey that covers a distance of 9.5 light-years, as measured by observers on Earth.

   (a)

   For the first propulsion stage, calculate the speed of the probe when the engines cut off and the distance it has travelled during this stage. The probe then coasts at this constant velocity for a further 2.0 × 10⁸ m before a course correction. Calculate the total time taken for both the accelerating stage and this coasting stage, and hence the average speed of the probe over the combined distance of both stages.

   [6 marks]

   (b)

   For the final relativistic leg of the journey, calculate the Lorentz factor of the probe, the time for this leg as measured on Earth, and the time for this leg as measured by a clock on the probe. Explain, in terms of proper time, why these two time intervals are different.

   [6 marks]

   Total for question 8: 12 marks

End of questions