Question 1                                                                                                             16 marks

(a) A particle is travelling along a flat frictionless surface with a velocity described by the

equation 2l(^) + 3t2J(^) . Initially at t = 0, it has a position of (0)0).

(i)   Calculate the speed of the particle at time t  = 3.0 s.

(ii)  Calculate, using (l(^)) J(^)) notation, the acceleration of the particle at t  = 3.0 s.

(iii) Calculate, using (l(^)) J(^)) notation, the position of the particle at time t  = 3.0 s.

(b) A 3.0 kg object has the following two forces acting on it:

If the object is initially at rest, determine its velocity v(→) at t  = 3.0 S.

[(3 + 3 + 4) + 5 = 15 marks]

Question 3 14 marks

(a) The figure shows the potential energy diagram for a 50 g particle that is released from rest at x  = 0.5 m.

(i)   Will the particle move in the positive or negative x-direction? Explain your answer.

(ii)  What is the maximum speed of the particle?

(iii) What is the speed of the particle at x  = 3.5 m?

(iv) At what position or positions will it change direction?

(v)  At what position or positions on its trajectory will the net force on the particle be zero?

(b) A  dark   matter  particle  of   mass mD    and velocity   VD      collides   elastically   with   a helium nucleus mHe  (at rest initially) whose mass is four times that of the dark matter particle. The helium nucleus is observed to move off at an angle θD(I) with velocity VID, and the dark matter particle at θIHe  with velocity VH(I)e  (As shown in the diagram).

(i)          By applying the relevant

conservation laws, write down 3 equations describing this collision in terms of variables mD, mHe, VD, VID, VIHe , θID, θIHe  and explain what conditions are required.

(ii)         In a particular situation, the dark matter particle’s initial speed is

2.2 × 105  m/s, θID  = 75.96o and  θIHe   = 45o. Determine the speeds of the two particles, VD(I) and VH(I)e  after the collision.

[(2 + 2 + 2 + 1 + 1) + (3 + 3) = 14 marks]

Four small spheres, each of mass M kg, are placed at the vertices of a square of side-length L metres. They are fixed in this configuration by a rigid framework of negligible mass.

(a)  What is the moment of inertia of this system with respect to rotation around an axis that is perpendicular to the plane of the square array and which passes through the centre of mass?

(b) What is the moment of inertia of this system with respect to rotation around an axis that passes through two of the masses that are diagonally opposite  one another on the square plane?

(c)  Use the parallel axis theorem and your answer to Part (a) to determine the

moment of inertia of the system with respect to rotation around an axis that is perpendicular to the plane of the square array and that passes through the midpoint of one of its edges.

(d)  If the angular momentum for each of the rotational axes above is the same, which one of (a), (b) or (c) would have the

(i)   greatest angular speed?

(ii)  greatest kinetic energy?

[3 + 3 + 5 + (2 + 2) = 15 marks]

[ 3 + 5 + 3 + 4 = 15 marks]

(c) The average lifetime of a charged π meson (rest mass 139.57 MeV/c2  or 2.49x10-28 kg) in its own frame of reference is 26.0 ns.

(i) Define proper time.

(ii) If the π meson moves with speed 0.95c with respect to the Earth, what is its lifetime as measured by an observer at rest on Earth?

(iii) What is the average distance it travels before decaying as measured by an observer at rest on Earth?

(d) A train, length d, is moving in the direction shown at velocity v, with respect to the tracks.

As the centre of the train passes stationary observer O, beside the train tracks, the observer sees two lightning bolts strike simultaneously at each end of the train, A’ and B’.

(i) What is the length of the train as seen by the stationary observer O?

(ii) Explain the relativity of simultaneity.

(iii) Show that the observer O’, sitting on the train at the midpoint between A’ and B’ will see the lightning strike at A’ before B’ .

(iv) What is the time difference between the lightning strikes as observed by O’ on the train?

[ 4 + 2 + (1 + 2 + 2) + (2 + 1 + 3 + 3) = 20 marks]

Question 7 9 marks

An astronaut finds herself on an unknown planet and needs to calculate the acceleration due to gravity on the planet. Using a piece of string 1.5 m long and a mass of 50 g as a pendulum, she  counts 10 complete oscillations in 40 seconds.

(a) Show that the acceleration due to gravity on the surface of the planet is 3.7 ms-2

With various observations (and a bit of trigonometry) the astronaut estimates the radius of this unknown planet to be 5,000 km.

(b) Equating Newton’s second law with Newton’s law of gravitation show:

(c) Using energy balance show that the escape speed is given by:

(d) Hence calculate the escape speed the astronaut will need to achieve to leave the planet.

[4 + 2 + 2 + 1 = 9 marks]

(i) Determine the period of oscillation of the spring-mass system.

(ii) Determine the amplitude of the oscillation.

[1 + 1 + 1 + 2 + 2) + (3 + 1 + 2) + (2 + 2) = 17 marks]

Question 9 19 marks

(a) Calculate the refractive index of a material in which the speed of light is 2.012 x 108  ms-1

(b) A coin is placed at the bottom of a beaker under 15cm of water. The refractive index of water is 1.333 (~4/3). Calculate the apparent depth of the coin.

(Assume the small angle approximation tanθ ≈ sinθ)

(d) What is the focal length of a 2.0 dioptre lens used in a pair of reading glasses?

(e) If the power of a typical human cornea (eye lens) is 40 dioptre, what is the combined focal length of the eye and the reading glasses in part (d) together?

(f) The diagram shows the object and image typical of a camera.

(i) How far from the camera lens must the detector be, if the lens focal length is 50 mm, to focus on an object 75 cm away?

(ii) Is the image real or virtual?

(iii) What is the magnification of the image?

[ 1 + 4 + (3 + 2) + 1 + 2 + (3 + 2 + 1) = 19 marks]

(c)  Briefly explain the Doppler effect in the context of sound waves.

(d)  If the frequency of a stationary emergency vehicle siren is 1 kHz, what is the difference   in frequency between that heard when the vehicle is approaching an observer at 20 m/s and the frequency heard as the vehicle moves away at the same speed?

[ 2 + 2 + 2 + 4 = 10 marks]