Consider a frictionless track as shown in Figure P6.48. A block of mass m1 = 4.30 kg is released from A. It makes a head on elastic collision at B with a block of mass m2 = 9.00 kg that is initially at rest. Calculate the maximum height to which m1 rises after the collision.
Do this in three steps. I would need to see your figure to provide numerical values. I assume m1 initially slides down some ramp a certain distance.
(1) Use conservation of energy to get the speed of m1 just before impact
(2) Apply cnservation of energy and momentum to the collision to get the final velocity of m1. Presumably it will recoil somewhat.
(3) From the recoil velocity, and conservation of energy, get the height m1 travels back up the track.
To calculate the maximum height to which m1 rises after the collision, we can use the principles of conservation of momentum and conservation of mechanical energy.
1. Conservation of Momentum:
Before the collision, the momentum of m1 is solely in the downward direction, as it is released from A. After the collision, the momentum of m1 changes direction and becomes solely upward due to the collision. However, the total momentum of the system (m1 + m2) remains constant.
Momentum before collision = Momentum after collision
m1 * initial velocity = m1 * final velocity1 + m2 * final velocity2
Since m2 is initially at rest, the equation becomes:
m1 * initial velocity = m1 * final velocity1
2. Conservation of Mechanical Energy:
In an elastic collision, the total mechanical energy of the system remains constant before and after the collision. This means that the initial potential energy of m1 is equal to the final potential energy when it reaches the maximum height.
Initial potential energy = Final potential energy
m1 * g * h1 = m1 * g * h_max
where g is the acceleration due to gravity, h1 is the initial height, and h_max is the maximum height reached by m1 after the collision.
Solving the equations:
From the first equation, we can find the final velocity of m1:
final velocity1 = initial velocity * (m1 - m2) / (m1 + m2)
Substituting this value into the second equation, we get:
m1 * g * h1 = m1 * g * h_max
Canceling out m1 and g, we get:
h1 = h_max
Therefore, the maximum height to which m1 rises after the collision is equal to the initial height, h1.
Note: It's important to ensure that the masses and velocities are in consistent units before plugging them into the equations.