Exercises: Conservation of momentum in two dimensions

A ball of mass $0.50\ \text{kg}$ moves with velocity components $v_x = 4.0\ \text{m s}^{-1}$ and $v_y = 3.0\ \text{m s}^{-1}$.

1. a) Calculate the speed of the ball.
2. b) Calculate the $x$ and $y$ components of its momentum.
3. c) Calculate the magnitude of its momentum.

Two snooker balls on a table in a club in Navan have equal masses of $0.16\ \text{kg}$. A ball moving at $2.0\ \text{m s}^{-1}$ along the $x$-axis strikes a stationary ball. After the collision the first ball moves at $1.2\ \text{m s}^{-1}$ at $+53^\circ$ to the $x$-axis, and the second moves off at an angle below the $x$-axis.

1. a) Calculate the total momentum in the $x$-direction before the collision.
2. b) State the total momentum in the $y$-direction before the collision.
3. c) State what the total $y$-momentum must be after the collision, and explain why the second ball must move below the axis.

A $1,000\ \text{kg}$ car travelling east at $10\ \text{m s}^{-1}$ and a $1,500\ \text{kg}$ van travelling north at $8.0\ \text{m s}^{-1}$ collide at a junction in Mullingar.

1. a) Calculate the momentum of the car and of the van, giving the direction of each.
2. b) Calculate the $x$ and $y$ components of the total momentum before the collision, taking east as $+x$ and north as $+y$.
3. c) State the principle you used to describe the total momentum after the collision, and the condition under which it applies.

A $1,000\ \text{kg}$ car travelling east at $10\ \text{m s}^{-1}$ and a $1,500\ \text{kg}$ van travelling north at $8.0\ \text{m s}^{-1}$ collide at a junction in Mullingar and stick together.

1. a) Calculate the components of the velocity of the wreckage immediately after the collision.
2. b) Calculate the speed of the wreckage and its direction, measured north of east.
3. c) Calculate the kinetic energy lost in the collision.

A $0.40\ \text{kg}$ puck on an air table moves at $3.0\ \text{m s}^{-1}$ in the $+x$ direction and strikes a stationary $0.60\ \text{kg}$ puck. After the collision the $0.40\ \text{kg}$ puck moves at $1.0\ \text{m s}^{-1}$ at $60^\circ$ above the $+x$ axis.

1. a) Calculate the $x$ and $y$ components of the velocity of the $0.60\ \text{kg}$ puck after the collision.
2. b) Calculate the speed and direction of the $0.60\ \text{kg}$ puck.
3. c) Determine whether the collision is elastic by calculating the kinetic energy before and after.

A stationary radioactive nucleus of mass $3.0\times10^{-25}\ \text{kg}$ decays and emits two particles. An alpha particle of mass $6.6\times10^{-27}\ \text{kg}$ moves off at $1.5\times10^{6}\ \text{m s}^{-1}$ along the $+x$ axis. A second particle of mass $1.0\times10^{-26}\ \text{kg}$ moves off along the $+y$ axis at $2.0\times10^{6}\ \text{m s}^{-1}$. The remaining daughter nucleus recoils.

1. a) State the total momentum before the decay.
2. b) Calculate the components of momentum of the daughter nucleus.
3. c) Calculate the magnitude of its momentum and the direction of its recoil relative to the $+x$ axis.

In an investigation of an elastic collision on a frictionless surface, a puck of mass $m$ moving at speed $u$ strikes a stationary puck of identical mass $m$. After the collision the pucks move off at angles $\theta_1$ and $\theta_2$ on opposite sides of the original line of motion.

1. a) Using conservation of momentum, write equations for the $x$ and $y$ directions.
2. b) By using conservation of kinetic energy and the vector triangle of momenta, show that $\theta_1+\theta_2=90^\circ$.
3. c) In one trial $u = 2.0\ \text{m s}^{-1}$ and $\theta_1 = 30^\circ$. Calculate the speeds of both pucks after the collision.
4. d) A student measures $\theta_1 = 30^\circ$ and $\theta_2 = 55^\circ$. State what this suggests about the collision and give two possible reasons.

At a cross-roads outside Ennistymon, a car of mass $1,200\ \text{kg}$ travelling east collides with a van of mass $2,000\ \text{kg}$ travelling north. They lock together and skid along a straight line at $35^\circ$ east of north, a distance of $14\ \text{m}$ before stopping. The coefficient of kinetic friction between the tyres and the road is $0.60$. Take $g = 9.8\ \text{m s}^{-2}$.

1. a) Calculate the speed of the wreckage immediately after the collision.
2. b) Calculate the momentum components of the wreckage after the collision.
3. c) Calculate the initial speeds of the car and of the van.
4. d) The speed limit on the road is $80\ \text{km h}^{-1}$. State, with a calculation, whether the car was speeding.

A neutron of mass $m$ and speed $v$ collides elastically with a stationary nucleus of mass $4m$ and is scattered through $90^\circ$ from its original direction.

1. a) Write equations for conservation of momentum in the directions parallel and perpendicular to the original motion of the neutron.
2. b) Using conservation of kinetic energy, find the speed of the neutron after the collision, in terms of $v$.
3. c) Find the speed of the nucleus after the collision and the angle at which it recoils to the original direction.
4. d) Calculate the fraction of the neutron's initial kinetic energy transferred to the nucleus, and explain why light nuclei are better moderators.