2D - Momentum Question, Grade 12 Canadian Physics

[ccvispartan]

Homework Statement

2 cars collide at an intersection.
one car which has a mass of 1,400kg has a velocity of 45 km/h;
the other car has a mass of 1,300kg and has a velocity of 39 km/h[E]
The cars has an inelastic collision. What are their velocity after the collision

Homework Equations

Conservation of momentum: P total = P total prime

The Attempt at a Solution

I have tried on numerous occasions but have not achieved the correct answer.

The correct answer should be: "30 km/h [ 51 South of East.]

I got the correct magnitude of the velocity which is 30 km/h. Easy enough, but i didn't get the degrees required. Additional vector diagram is appreciated.

 

[grzz]

Try to draw the vector diagram for the initial total momentum of the system.

 

[ccvispartan]

@grzz I figured out what i did wrong. When i calculated for the theta of the vector triangle. I simply used the velocities. Instead i tried it with m(v) and it worked for me. Thanks.

 

[grzz]

You just confirmed that it is the total momentum that is conserved and not the total velocity!

 

[asteriatic]

I would approach this problem by first setting up a coordinate system, with the x-axis pointing east and the y-axis pointing north. I would then use the conservation of momentum equation, P total = P total prime, to solve for the final velocities of the two cars.

Let us assume that the car with a mass of 1,400kg is traveling in the positive x-direction (east) and the car with a mass of 1,300kg is traveling in the positive y-direction (north).

Using the given information, we can calculate the initial momentum of each car as follows:

P1 = m1v1 = (1,400kg)(45 km/h) = 63,000 kg*km/h [E]
P2 = m2v2 = (1,300kg)(39 km/h) = 50,700 kg*km/h [N]

Since this is an inelastic collision, the two cars will stick together after the collision and move in the same direction. Therefore, the final momentum of the combined cars will be:

P total prime = (m1 + m2)v final

Using the conservation of momentum equation, we can equate the initial momentum to the final momentum and solve for the final velocity:

P total = P total prime
63,000 kg*km/h [E] + 50,700 kg*km/h [N] = (1,400kg + 1,300kg)v final
113,700 kg*km/h [E] = 2,700kg v final

Dividing both sides by 2,700kg, we get:

v final = 42 km/h [E]

Therefore, the final velocity of the combined cars after the collision will be 42 km/h [E].

To find the direction of the final velocity, we can use the tangent function:

tan θ = (v final y)/(v final x)
tan θ = (42 km/h)/(42 km/h)
θ = tan^-1(1)
θ = 45°

Since the final velocity is in the east direction (positive x-axis), the angle of 45° indicates that the final velocity is moving 45° south of east.

Therefore, the final velocity of the combined cars after the collision is 42 km/h [45° South of East].