The Italian Job

Day 1: Cars, Gold, and Motion

Problem 1

In The Italian Job, the Mini Coopers must carry stolen gold while still remaining agile.

Suppose one Mini Cooper has mass \(m_c = 1.0\times 10^3\ {\rm kg}\) and carries an additional gold mass of \(m_g = 320\ {\rm kg}\).

(a) What is the total mass of the loaded car?

(b) If the loaded car accelerates from rest at \(a = 2.8\ {\rm m/s^2}\), what net force is required?

(c) Compare this to the net force required for the unloaded car at the same acceleration.

What does this suggest about the effect of the gold on the car’s performance?

Problem 2

A loaded Mini Cooper rounds a corner at speed \(v = 16\ {\rm m/s}\) on a flat road of radius \(r = 32\ {\rm m}\).

Use the loaded mass from Problem 1.

(a) What centripetal acceleration is required?

(b) What centripetal force is required?

(c) What minimum coefficient of static friction is needed to make the turn without skidding?

Problem 3

At one point a car must climb a steep parking ramp while carrying the gold.

Suppose the ramp makes an angle of \(\theta = 14^\circ\) with the horizontal. The loaded car has mass \(m = 1.32\times 10^3\ {\rm kg}\).

(a) What component of the car’s weight acts down the ramp?

(b) What normal force acts on the car?

(c) If the car climbs the ramp at constant speed, what uphill driving force must the tires provide to balance the downhill component of the weight?

Neglect frictional losses other than the force needed to maintain traction.

Day 2: Braking, Impulse, and Power

Problem 4

A Mini Cooper brakes to avoid hitting an obstacle.

Suppose the loaded car is traveling at \(v_i = 20\ {\rm m/s}\) and comes to rest in \(\Delta t = 3.5\ {\rm s}\).

(a) What is the acceleration of the car?

(b) What average braking force acts on the car?

(c) How far does the car travel while braking?

Use the loaded mass \(m = 1.32\times 10^3\ {\rm kg}\) and assume constant acceleration.

Problem 5

Suppose one of the loaded cars collides softly with a barrier and comes to rest from \(v = 12\ {\rm m/s}\) in \(\Delta t = 0.18\ {\rm s}\).

(a) What is the change in momentum of the car?

(b) What is the average force on the car during the collision?

Use the loaded mass \(m = 1.32\times 10^3\ {\rm kg}\).

Problem 6

A car climbing a hill at speed \(v = 12\ {\rm m/s}\) must overcome the component of gravity along the ramp.

Using the ramp in Problem 3 and the loaded mass \(m = 1.32\times 10^3\ {\rm kg}\):

(a) What power is required just to overcome gravity while maintaining constant speed?

(b) If rolling and air-resistance losses add an additional \(4.0\times 10^3\ {\rm W}\), what total power is required?

Assume the speed is constant.