Question

A car with mass mc=1370kg is traveling west through an intersection with a speed of vc=13.71m/s when a truck of mass mt=1840kg traveling south at vt=12.83m/s fails to yield. They collide, stick together, and slide on the asphalt with a coefficient of kinetic friction of μk=0.500. What will be their final velocity after the collision?

Answer 1

To determine the final velocity after a collision, use conservation of momentum for the west and south directions independently, solve for the velocity components, and combine them to find the total velocity vector.

Step-by-step explanation:

To find the final velocity after the collision of the car and the truck, we need to use the principle of conservation of momentum. As they collide and stick together, their total momentum just before the collision is equal to their total momentum just after the collision. We have to consider momentum in two perpendicular directions, west and south, since they are traveling in those directions respectively.

The initial momentum of the car (mc * vc) in the west direction is (1370 kg * 13.71 m/s), and the initial momentum of the truck (mt * vt) in the south direction is (1840 kg * 12.83 m/s). After the collision, the combined mass (mc + mt) will move with a certain velocity (v) at some angle. Let's call the components of this velocity in the west and south directions vw and vs, respectively.

Conserving momentum in the west direction:

mc * vc = (mc + mt) * vw

Conserving momentum in the south direction:

mt * vt = (mc + mt) * vs

The final velocity vector can be found by combining these two components using the Pythagorean theorem:

v = sqrt(vw^2 + vs^2)

Now, the effect of kinetic friction becomes only relevant if we are asked about the distance covered after the collision until the cars come to a stop, which is not the case here. Therefore, the kinetic friction does not affect the calculation of the final velocity immediately after the collision.

Answer 2

The final velocity of the combined car and truck immediately after the collision is approximately
`\(9.40 \, \text{m/s}\).`

To solve this problem, we can use the principles of conservation of momentum and the physics of kinetic friction. Let's break it down into steps:

Step 1: Conservation of Momentum

The law of conservation of momentum states that the total momentum of a system remains constant if no external forces act on it. In this case, the system consists of the car and the truck before and after the collision. Since they stick together after the collision, we can consider them as a single object post-collision.

Initial Momentums:

- Momentum of the car
`(mc * vc): \( m_c * v_c = 1370 \, \text{kg} * 13.71 \, \text{m/s} \)`

- Momentum of the truck
`(mt * vt): \( m_t * v_t = 1840 \, \text{kg} * 12.83 \, \text{m/s} \)`

Since the car is moving west and the truck is moving south, their momentums are perpendicular to each other. We need to find the resultant momentum vector using Pythagoras' theorem.

Resultant Initial Momentum:

`\( p = √((m_c v_c)^2 + (m_t v_t)^2) \)`

Step 2: Post-Collision Velocity

After the collision, since the car and truck stick together, their combined mass is
`\( m_c + m_t \).` The final velocity
`(\( v_f \))` can be found by dividing the resultant initial momentum by the total mass.

`\[ v_f = (p)/(m_c + m_t) \]`

Step 3: Kinetic Friction

After the collision, the combined object will slide across the asphalt, experiencing kinetic friction. This will eventually bring the object to a stop. However, the final velocity immediately after the collision is not affected by friction. Friction will only determine how far they slide before stopping.

Calculations

Let's calculate the final velocity immediately after the collision using these steps.

First, calculate the initial momentums of the car and the truck, then their resultant momentum, and finally the post-collision velocity.

The final velocity of the combined car and truck immediately after the collision is approximately
`\(9.40 \, \text{m/s}\).`

This velocity is the speed at which the car and truck, now stuck together, will move immediately after the collision. The direction of this motion will be along the line of the resultant momentum vector, which can be calculated using the angle formed with respect to one of the initial velocity directions.

However, it's important to note that this is the velocity right after the collision. Due to the kinetic friction acting on the asphalt, they will gradually slow down and eventually come to a stop. The calculation of the distance they slide before stopping would require additional steps involving the work-energy principle.