Consider a frictionless track as shown in the figure below. A block of mass m1 = 4.25 kg is released from A. It makes a head on elastic collision at B with a block of mass m2 = 10.0 kg that is initially at rest. Calculate the maximum height to which m1 rises after the collision.
Kinetic energy and momentum are both conserved in the collision. The two equations allow you to solve for the two velocities after collision. Assume motion of both masses is along one axis only (forward or backward), since the collision is head-on.
I will leave that exercise for you to complete. Since no figure was provided, I may end up solving the wrong situation.
If m1 is free to move upwards after collision, either up a frictionless ramp or on a pendulum string, its kinetic energy after collision will be converted to potential energy.
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To calculate the maximum height to which m1 rises after the collision, you can use the principle of conservation of mechanical energy.
Here's how you can do it step by step:
1. Calculate the initial kinetic energy of block m1 just before the collision:
- The initial velocity of m1 can be calculated using the conservation of linear momentum:
m1 * v1 = m2 * v2, where v1 is the velocity of block m1 just before the collision and v2 is the velocity of block m2 just before the collision.
Since m1 is released from rest, v1 = 0.
From the equation, v2 = (m1/m2) * v1 = 0, as m2 is initially at rest.
- Therefore, the initial kinetic energy of m1 is 0.5 * m1 * v1^2 = 0.
2. Calculate the final velocities of both blocks after the collision using the conservation of linear momentum:
- The momentum before the collision is equal to the momentum after the collision:
m1 * v1 = m1' * v1' + m2' * v2', where v1' and v2' are the final velocities of blocks m1 and m2 respectively.
Since the collision is elastic, the total kinetic energy after the collision is equal to the total kinetic energy before the collision. Therefore:
0.5 * m1 * v1^2 = 0.5 * m1' * v1'^2 + 0.5 * m2' * v2'^2.
- Since block m2 is initially at rest, v2' = 0.
- Solve the system of equations to find the final velocities of both blocks.
3. Calculate the maximum height to which m1 rises using energy conservation:
- The potential energy gained by m1 when it reaches the maximum height is given by m1 * g * h_max, where g is the acceleration due to gravity and h_max is the maximum height.
- The final kinetic energy of m1 at the maximum height is given by 0.5 * m1 * v1'^2, as it is at rest.
- Therefore, equating the potential energy gained to the final kinetic energy:
m1 * g * h_max = 0.5 * m1 * v1'^2.
- Solve for h_max to get the maximum height.
By following these steps and plugging in the given values of m1, m2, and g into the equations, you should be able to calculate the maximum height to which m1 rises after the collision.