Question

As shown in the figure below, we have three air carts (labeled #1, #2, and #3) on an air track. The carts are equipped with spring bumpers so that all collisions are elastic. The masses of the carts are
m1 = 5m,

m2 = 2m,
and
m3 = m.
Initially, cart #1 is given a velocity to the right of magnitude
v0
and the other carts are at rest. Determine the following.

(a) final speed of each cart (Indicate the direction with the sign of your answer.)

(b) ratio of the total kinetic energy before any collision to the total kinetic energy after all collisions

I know that they are elastic collisions and I have solved for the final velocity of the first cart, but I'm not sure where to go from there.

Answer 1

To solve this problem, we can break it down into two elastic collisions (first between carts #1 and #2 and then between carts #2 and #3). We are given the initial velocities of the three carts as:

v1_initial = v0
v2_initial = 0
v3_initial = 0

For the first collision between carts #1 and #2, we can use the equations for elastic collisions:

v1_final = (m1 - m2) / (m1 + m2) * v1_initial + 2 * m2 / (m1 + m2) * v2_initial
v2_final = 2 * m1 / (m1 + m2) * v1_initial - (m1 - m2) / (m1 + m2) * v2_initial

Substitute the given values for the masses and initial velocities:

v1_final = (5m - 2m) / (5m + 2m) * v0 + 2 * 2m / (5m + 2m) * 0
v1_final = 3m / 7m * v0
v1_final = 3/7 * v0

v2_final = 2 * 5m / (5m + 2m) * v0 - (5m - 2m) / (5m + 2m) * 0
v2_final = 10m / 7m * v0
v2_final = 10/7 * v0

Now, we have the final velocities of carts #1 and #2 after the first collision. Moving on to the second collision between carts #2 and #3, remember that cart #1 is no longer involved and its final velocity remains the same:

v1_final = 3/7 * v0

For the second collision between carts #2 and #3, use the same elastic collision equations:

v2_final = (m2 - m3) / (m2 + m3) * v2_initial + 2 * m3 / (m2 + m3) * v3_initial
v3_final = 2 * m2 / (m2 + m3) * v2_initial - (m2 - m3) / (m2 + m3) * v3_initial

Substitute the new values for the masses and initial velocities (using the final velocities from the previous collision):

v2_final = (2m - m) / (2m + m) * (10/7 * v0) + 2 * m / (2m + m) * 0
v2_final = 4/7 * v0

v3_final = 2 * 2m / (2m + m) * (10/7 * v0) - (2m - m) / (2m + m) * 0
v3_final = 20/7 * v0

So, the final speeds of the three carts are:
Cart #1: 3/7 * v0 (to the right)
Cart #2: 4/7 * v0 (to the right)
Cart #3: 20/7 * v0 (to the right)

(b) The ratio of the total kinetic energy before any collision to the total kinetic energy after all collisions can be determined as follows:

initial_KE = 1/2 * m1 * v1_initial^2 + 1/2 * m2 * v2_initial^2 + 1/2 * m3 * v3_initial^2
initial_KE = 1/2 * 5m * v0^2

final_KE = 1/2 * m1 * v1_final^2 + 1/2 * m2 * v2_final^2 + 1/2 * m3 * v3_final^2
final_KE = 1/2 * 5m * (3/7 * v0)^2 + 1/2 * 2m * (4/7 * v0)^2 + 1/2 * m * (20/7 * v0)^2

Solve the final_KE equation and simplify:

final_KE = 1/2 * 5m * 9/49 * v0^2 + 1/2 * 2m * 16/49 * v0^2 + 1/2 * m * 400/49 * v0^2
final_KE = 45/98 * m * v0^2 + 16/49 * m * v0^2 + 200/49 * m * v0^2

Now, calculate the ratio:

KE_ratio = initial_KE / final_KE
KE_ratio = (1/2 * 5m * v0^2) / (61/98 * m * v0^2)

Simplify the equation by canceling out the mass and velocity squared terms:

KE_ratio = (1/2 * 5) / (61/98)
KE_ratio = 98/61

Thus, the ratio of the total kinetic energy before any collision to the total kinetic energy after all collisions is 98/61.