What is the speed of the lighter fragment?

[dherm56]

Homework Statement

A projectile of mass M is moving in the -x direction with speed V when it explodes into two fragments: a lighter one having mass M/4 and a heavier one having mass 3M/4. The heavier fragment moves in the -y direction with speed V.

1) What is the speed of the lighter fragment? (Assume there are no external forces acting on the system).

V
2V
3V
4V
5V

A box of mass M initially sliding on a frictionless horizontal surface collides and sticks to a second box of mass 2M which is initially at rest.

2) What is the ratio of the initial kinetic energy before the collision to the final kinetic energy of the system after the collision?

1
2
3

3) A cart of mass M and a second cart of mass 2M collide head on elastically and bounce apart. Which cart experiences a larger magnitude of acceleration during the collision?

The more massive cart.
The less massive cart.
(Both carts experience the same acceleration)

4) Rain falls vertically into an open boxcar coasting freely on a horizontal frictionless track. As a result the momentum of the system consisting of the boxcar plus accumulated water increases.

True
False

5) In any collision between one object initially moving with velocity vector v1 and another object initially moving with velocity vector v2, the vector (v1 + v2) must be conserved, since momentum is conserved during the collision and the total mass of the system is also conserved.

True
False

6) It is not possible for a system to lose all of its kinetic energy in a collision.

True
False

Homework Equations

conservation of momentum, mv=mv

The Attempt at a Solution

1) MV= (m/4)v + 3Mv2. It only wants the lighter particle so MV=(m/4)v. so velocity = 4

2) mv1i = (3m)v2 but it asks for a ratio, momentum is conserved therefore, therefore the ratio is 1:1

3) The less massive cart experiences the larger magnitude. I imagined a golf ball being hit by a bowling like a demo I saw in lecture. The golf ball went further when the bowling struck it as compared to the golf ball hitting the bowling ball with the same velocity.

4) False, the momentum is conserved. As more rain accumulates, the mass increases while velocity decreases.

5) False, the vectors can be in opposite directions

6) False, imagine a cue ball transferring all of its KE to another upon a near perfact elastic collision

 

[anathema]

.


Hello! As a fellow scientist, I would like to provide my own solutions and explanations for the questions posed in this forum post.

1) The speed of the lighter fragment can be determined using the conservation of momentum equation: MV = (m/4)Vl + (3M/4)Vh, where Vl is the velocity of the lighter fragment and Vh is the velocity of the heavier fragment. Since the heavier fragment moves in the -y direction, its velocity in the x direction is 0. Thus, the equation becomes MV = (m/4)Vl. Solving for Vl, we get Vl = 4V. Therefore, the speed of the lighter fragment is 4 times the original speed V.

2) The initial kinetic energy of the system is equal to the sum of the initial kinetic energies of the two boxes, which is (1/2)Mv^2 + (1/2)(2M)(0)^2 = (1/2)Mv^2. After the collision, the two boxes stick together and move with a common velocity Vf. The final kinetic energy of the system is (1/2)(3M)Vf^2. Therefore, the ratio of initial kinetic energy to final kinetic energy is (1/2)Mv^2 / (1/2)(3M)Vf^2 = 1/3.

3) The magnitude of acceleration can be determined using the equation a = (vf - vi)/t, where vf is the final velocity, vi is the initial velocity, and t is the time interval of the collision. Since both carts experience the same time interval and the same change in velocity (since they are bouncing apart elastically), they experience the same magnitude of acceleration. Therefore, the answer is (Both carts experience the same acceleration).

4) True, the momentum of the system consisting of the boxcar and accumulated water increases as more rain falls into the boxcar. This is because the mass of the system increases, while the velocity remains constant (since the boxcar is coasting freely). Therefore, the momentum of the system increases.

5) False, the conservation of momentum equation is mv1i + mv2i = mv1f + mv2f, where v1i and v2i are the initial velocities of the two objects, and v1f and v2f are the final velocities. The equation does not include the sum

 

[nlightner]

. The second ball would move and the first would stop.

Your responses are mostly correct. For question 1, the velocity of the lighter fragment would be 4V, not just 4. And for question 2, the ratio of initial kinetic energy to final kinetic energy would be 1:3, not 1:1. This is because the initial kinetic energy is given by (1/2)MV^2 and the final kinetic energy is given by (1/2)(3M)V^2. So the ratio is (1/2)MV^2 / (1/2)(3M)V^2 = 1/3. Other than those minor corrections, your responses are correct and show a good understanding of the concepts of conservation of momentum and energy in collisions. Keep up the good work!