A railroad car that weighs 20,000 lb is traveling eastward with a

velocity whose magnitude is 5fps. A second car, on the same track, that
weighs 40,000 lbs also traveling in an easterly direction with a
velocity of 7.8fps. When the cars collided, they became coupled together
and then both cars moved with the same velocity. If friction between the
cars and the rails is neglected, what were the magnitude and the
direction of the velocity of the car after the collision?

Momentum before = m1 v1 + m 2 v2

Momentum after = (m1+m2) v

then
momentum after = momentum before.

To determine the magnitude and direction of the velocity of the car after the collision, we can use the principle of conservation of momentum. According to this principle, the total momentum before the collision should be equal to the total momentum after the collision.

The momentum of an object is the product of its mass and velocity. In this case, we have two objects, each with its own mass and velocity. Let's denote the mass of the first car as m1 and its velocity as v1, and similarly, the mass of the second car as m2 and its velocity as v2.

Before the collision, the total momentum is given by:

Total momentum before = m1 * v1 + m2 * v2

After the collision, the two cars become coupled together and move with the same velocity. Let's denote this common velocity as v.

After the collision, the total momentum is given by:

Total momentum after = (m1 + m2) * v

According to the conservation of momentum principle, these two total momenta should be equal. Therefore, we can set them equal to each other and solve for v:

m1 * v1 + m2 * v2 = (m1 + m2) * v

Now we can substitute the given values into this equation. The mass of the first car (m1) is 20,000 lb, the mass of the second car (m2) is 40,000 lb, the velocity of the first car (v1) is 5 fps, and the velocity of the second car (v2) is 7.8 fps.

(20,000 lb) * (5 fps) + (40,000 lb) * (7.8 fps) = (20,000 lb + 40,000 lb) * v

Now we can solve this equation to find the value of v, which will give us the magnitude and direction of the velocity of the car after the collision.