Collisions?

Question

If a car weighing 1000 Kg travelling East at 24 m/s collides with a truck weighing 3000 Kg traveling North at 15 m/s, what is the magnitude and direction of the resulting momentum if they're stuck together?

That same car weighing 1000 Kg is traveling such that it strikes the same truck as above at a 60 degree angle, what is the resulting magnitude and direction momentum if they stick together?

Answer 1

For the first case:

Since total momentum before and after the collision conserve, we calculate the horizontal and vertical momentum before collision, add them up in a vector sense, will give you the total momentum after the collision.

Horizontal momentum: 1000(24) = 24000 kg m/s
Vertical momentum: 3000(15) = 45000 kg m/s
(Final momentum)^2 = 24000^2 + 45000^2 (because is right angle)
Final momentum = 51000 kg m/s

To solve the angle, we use trigs,

tan(alpha) = 24000/45000
alpha = 28.07 deg
or N 28.07 deg E


For the second case, since it is not in right angle, you have to use cosine law to solve, the two vectors have the same magnitude as above, the angle between them is 60 deg. However, when you add these two vectors up, watch out for the vector direction. You have to add them by head to tail. Draw out the vectors, solve the angle using common angle with parallel lines situation, you will find that the angle between the two vectors be 120 deg.

(Final momentum)^2 = 45000^2 + 24000^2 - (45000)(24000)cos(120)

Final momentum = 60671.245 kg m/s
(you should realize this momentum is slightly higher than the one above as you may realized the resultant vector in this case is longer).

To solve the angle, you can use sin law,

sin (beta) / 24000 = sin (120) / 60671.245
beta = 20.03 deg

However, to find the angle from E to the resultant angle, you should first find the angle from E to the 45000 vector, which is 60 deg. Take away the 20.03 deg (which is between the 45000 vector and the resultant vector), you get 39.966 deg.
Therefore, the final direction is E 39.966 deg N.

Answer 2

Conservation of momentum. The East and North momentums must remain the same.