Question

in a certain road accident, a car of mass 2000 kg, traveling south, collided in the middle of an intersection with a truck of mass 6000 kg, traveling west. the vehicles locked and skidded off the road along a line pointing almost exactly southwest. a witness claimed that the truck had entered the intersection at 50 mph. a) if the witness is telling the truth, how fast was the car moving before the collision? does this seem likely?

Answer 1

By applying the conservation of momentum and assuming equal momentum components post-collision, we calculate that the car must have been traveling approximately 150 mph to satisfy the conditions of the collision, which does not seem likely for typical traffic conditions.

Step-by-step explanation:

To determine how fast the car was moving before the collision, we can use the principle of conservation of momentum in a two-dimensional collision. Since the truck was traveling west and the car was traveling south, and after the collision, the vehicles moved southwest, the momentum vectors added up to a southwest direction.

In the context of the question, we can define south as the negative y-direction and west as the negative x-direction. The witness claims the truck (6000 kg) was traveling at 50 mph west, which we need to convert to meters per second (m/s) before performing calculations. The conversion factor is 1 mph ≈ 0.44704 m/s. Therefore, the velocity of the truck is 50 mph * 0.44704 m/s/mph ≈ 22.352 m/s.

The conservation of momentum states that the momentum before the collision is equal to the momentum after the collision. The momentum of the system post-collision can be broken down into its x and y components, which should be equal due to the fact that the final direction is southwest (45 degrees off of both axes).

The mass of the car is 2000 kg. Let v be the speed of the car south before the collision. The momentum in the west direction (-x) is entirely due to the truck: (6000 kg)(-22.352 m/s), and the momentum in the south direction (-y) will be due to the car: (2000 kg)(v).

The magnitudes of the x and y components after the collision should be equal, leading to the following equation:
6000 kg * 22.352 m/s = 2000 kg * v

We can solve for v, the car's velocity:
v = (6000 kg * 22.352 m/s) / 2000 kg
v ≈ 67.056 m/s

If we convert 67.056 m/s back to mph (1 m/s ≈ 2.23694 mph), we find the speed of the car is approximately 150 mph. This is an unusually high speed for a car to be traveling at an intersection and does not seem likely, suggesting that the witness' claim may need to be further investigated.