Momentum in Two Dimensions
Practice
practice problem 1
Your consulting agency, with its extensive knowledge of physics, has been subpoenaed to provide expert testimony at an automobile accident case in civil court. The case involves a crash between a compact car, loaded with a driver and three passengers, and a tractor-trailer truck. Prior to your testimony you were provided with information from the local police department and the manufacturers of the car and truck.
- The police report stated that the force needed to drag a 130 N (29 lb) car tire across the pavement at a constant velocity at this location was no less than 100 N (22 lb). Specifications provided by the truck's manufacturer claim that, due to the manner in which the truck balances braking across its 3 axles, the effective coefficient of friction for truck tires is only 70% that of car tires. What is the coefficient of kinetic friction between the tires and the road for both the car and the truck?
- After colliding, the car and the truck skidded to a stop as shown in the diagram. The car skidded a distance of 8.2 m (27 ft) at 33° while the truck skidded 11 m (36 ft) at 7°. What was the speed of each vehicle immediately after the moment of impact?
- Before colliding, the car was traveling due north and the truck was traveling due west as shown in the diagram. What was the speed of each vehicle immediately before the moment of impact? The mass of the car is 1,380 kg (3,040 lb) and the mass of the truck is 7,120 kg (15,700 lb).
- At this intersection, the truck driver had a flashing yellow light while the car driver had a flashing red light. In situations like this, the driver with the flashing yellow is expected to be extra vigilant for cross traffic while the driver with the flashing red is expected to come to a complete stop and wait until the intersection is safe to enter.
- The truck driver claims that the car ran the flashing red light. He also claims to have begun braking in anticipation of a collision; traveling at only 6.7 m/s (15 mph) at the moment of impact.
- The car driver claims to have made a full stop at the light before entering the intersection. He also claims that the truck driver did not see him until after the collision.
- The police report stated that the distance from the traffic light to the collision point for the car was 14 m (46 ft).
- Specifications provided by the car's manufacturer claim that the maximum acceleration of a comparably loaded vehicle is about 3.0 m/s2 (10 ft/s2).
Adapted from Feldman, 1997
solution
Text
fk = μkN
Text
μcar = fminimum Wdrag tire μcar = 100 N 130 N Text
μtruck = nμcar μtruck = 0.70(0.7692) Text
K′ = W ½mv′2 = fΔs Text
fk = μkN = μmg
Text
½mv′2 = μmgΔs v′car = √(2μgΔs) Text
v′car = √(2μgΔs) v′car = √[2(0.7692)(9.8 m/s2)(8.2 m)] Repeat using values for the truck.
v′truck = √(2μgΔs) v′truck = √[2(0.5385)(9.8 m/s2)(11 m)] Text
pcar = p′y car + p′y truck mcarvcar = mcarv′car sin θcar + mtruckv′truck sin θtruck (1,380 kg)vcar = (1,380 kg)(11.12 m/s)sin(33°) + (7,120 kg)(10.77 m/s)sin(7°) (1,380 kg)vcar = 8,357 kg m/s + 9,349 kg m/s (1,380 kg)vcar = 17,706 kg m/s vcar = 17,706 kg m/s 1,380 kg ptruck = p′x car + p′x truck mtruckvtruck = mcarv′car cos θcar + mtruckv′truck cos θtruck (7,120 kg)vtruck = (1,380 kg)(11.12 m/s)cos(33°) + (7,120 kg)(10.77 m/s)cos(7°) (7,120 kg)vtruck = 12,869 kg m/s + 76,143 kg m/s (7,120 kg)vtruck = 89,012 kg m/s vtruck = 89,012 kg m/s 7,120 kg We've already done the work needed to answer the first half of this question. The speed we just calculated is much greater than the truck driver's claim.
12.50 m/s ≫ 6.7 m/s
It is unlikely the truck driver was braking before the collision and there is a good chance he did not see the car entering the intersection.
The second half of this question requires an additional calculation. What acceleration is needed for the car to travel from the stop light to the point of impact in the distance given and with the pre‑crash speed we calculated earlier?
v2 = v02 + 2aΔs
a = v2 2Δs a = (12.83 m/s)2 2(14 m) a = 5.879 m/s2 (19 ft/s2) This is much greater than the maximum acceleration given by the car's manufacturer.
5.879 m/s2 ≫ 3.0 m/s2
It is unlikely the car driver entered the intersection after coming to a complete stop at the traffic light. He probably ran the red light.
| car | truck | |
|---|---|---|
| vehicle mass |
1,380 kg | 7,120 kg |
| measured skid distance |
8.2 m | 11 m |
| measured skid direction |
33° | 7° |
| measured pre‑crash direction |
90° | 0° |
| drag tire weight |
130 N | n/a |
| measured friction force |
100 N | n/a |
| calculated coefficient of friction | 0.7692 | 0.5385 |
| reconstructed post‑crash speed |
11.12 m/s | 10.77 m/s |
| reconstructed pre‑crash speed |
12.83 m/s | 12.50 m/s |
| reported pre‑crash speed |
n/a | 6.7 m/s |
| maximum acceleration |
3.0 m/s2 | n/a |
| rest to impact distance |
14 m | n/a |
| required acceleration |
5.842 m/s² | n/a |
| expert opinion |
failure to stop for a flashing red light | driver inattention at a flashing yellow light |
practice problem 2
Write something else.
solution
Answer it.
practice problem 3
Write something different.
solution
Answer it.
practice problem 4
Write something completely different.
solution
Answer it.