The figure below shows the experimental setup to study the collision between two carts.
In the experiment cart A rolls to the right on the level track, away from the motion sensor at the left end of the track. Cart B is initially at rest. The mass of cart A is equal to the mass of cart B. Suppose the two carts stick together after the collision. Assume the carts move frictionlessly.
The kinetic energy of the two carts after the collision
1)is equal to one half the kinetic energy of cart A before the collision.
2)is equal to one quarter the kinetic energy of cart A before the collision.
3)is equal to the kinetic energy of cart A before the collision.
4)is equal to twice the kinetic energy of cart A before the collision.
5)is equal to four times the kinetic energy of cart A before the collision.
6)None of the above.
mv=(2m)u
u=v/2
KE(of two carts) =(2m)u²/2=mv²/4 =0.5(mv²/2) =0.5KE(cart A)
Ans 1)
To determine the kinetic energy of the two carts after the collision, we need to consider the principle of conservation of momentum.
According to the principle of conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision, assuming no external forces act on the system.
In this case, we can assume that the collision is an elastic collision because the carts stick together after the collision.
Since cart A is initially moving to the right on the level track and cart B is initially at rest, the momentum before the collision can be written as:
Initial momentum = (mass of cart A) * (velocity of cart A) + (mass of cart B) * (velocity of cart B)
= (mass of cart A) * (initial velocity of cart A) + 0 (since cart B is at rest initially)
= (mass of cart A) * (initial velocity of cart A)
After the collision, since the two carts stick together, their velocity will be the same. Let's call it V.
Hence, the total momentum after the collision can be written as:
Final momentum = (mass of cart A + mass of cart B) * V
= 2 * (mass of cart A) * V
Since momentum is conserved, we can set the initial and final momenta equal to each other:
(mass of cart A) * (initial velocity of cart A) = 2 * (mass of cart A) * V
Canceling out the mass of cart A, we have:
(initial velocity of cart A) = 2 * V
Hence, the velocity of the two carts after the collision is one-half of the initial velocity of cart A.
Now, let's consider the kinetic energy of the two carts after the collision. The kinetic energy is given by the formula:
Kinetic energy = (1/2) * (total mass) * (velocity)^2
Before the collision, the kinetic energy of cart A can be written as:
Initial kinetic energy of cart A = (1/2) * (mass of cart A) * (initial velocity of cart A)^2
After the collision, the kinetic energy of the two carts together can be written as:
Final kinetic energy = (1/2) * (total mass) * (velocity)^2
= (1/2) * (2 * mass of cart A) * (V)^2
= (mass of cart A) * (V)^2
Comparing the final kinetic energy with the initial kinetic energy of cart A, we have:
Final kinetic energy = (mass of cart A) * (V)^2 = Initial kinetic energy of cart A
Therefore, the correct answer is option 3) The kinetic energy of the two carts after the collision is equal to the kinetic energy of cart A before the collision.