Question

A car equipped with a massless spring in front collides with a stationary car as shown in the diagram. The mass of the incoming car is three times the mass of the target car. While the spring is being compressed, the cars are moving closer together, and while it is expanding they are moving farther apart, but at the instant that the spring is fully compressed, they have no relative motion, which means they are moving at the same speed. At this instant, the cars are moving with a speed of 2.4 m/s.
(a) Find the speed of the incoming car before the collision.
(b) Find the fraction of the system’s energy that is stored in the spring when
it is fully compressed.
(c) Find the final speed and direfction for both cars after the spring expands and the cars separate.

Answer 1

Answer:

Explanation:

Let m be target car mass

a)

conservation of momentum

(3m)u + m(0) = (3m + m)(2.4)

               3u = 4(2.4)

                u = 3.2 m/s

b)

system energy is originally in the striking car as kinetic energy

KE = ½(3m)3.2² = 15.36m J

at full compression, kinetic energy is

½(3m + m)2.4² = 11.52m J

so spring potential energy fraction of total energy is

(15.36 - 11.52)/15.36 = 0.25 or 25%

c)

The center of mass (CoM) is moving at 2.4 m/s

The CoM sees the larger mass car approach at

3.2 - 2.4 = 0.8 m/s and will see it depart at - 0.8 m/s

so a ground based observer would see it moving at

v₁ = 2.4 + -0.8 = 1.6 m/s in its original direction

The CoM sees the smaller mass car approach at

0 - 2.4 = -2.4 m/s and will see it depart at + 2.4 m/s

so a ground based observer would see it moving at

v₂ = 2.4 + 2.4 = 4.8 m/s in its original direction of the larger car

so the relative velocity of approach at 3.2 - 0 = 3.2 m/s

equals the relative velocity of departure at 4.8 - 1.6 = 3.2 m/s

checks.