Two blocks are free to slide along the frictionless wooden track ABC shown in Figure P9.20. A block of mass m1 = 5.00 kg is released from A. Protruding from its front end is the north pole of a strong magnet, repelling the north pole of an identical magnet embedded in the back end of the block of mass m2 = 10.0 kg, initially at rest. The two blocks never touch. Calculate the maximum height to which m1 rises after the elastic collision.
STUDENT
ADIVCE
To find the maximum height to which m1 rises after the elastic collision, we can use the conservation of energy principle.
Let's label the maximum height as h. At this height, the speed of m1 will momentarily become zero before it starts descending.
Step 1: Calculating the initial energy
The initial energy of the system is simply the potential energy of m1 when it is at its initial position A:
E_initial = m1 * g * h
Step 2: Calculating the final energy
At the maximum height h, the velocity of m1 becomes zero. The final energy is the potential energy of m1 at this height plus the kinetic energy of m2 just before the collision.
E_final = m1 * g * 0 + 0.5 * m2 * v^2
where v is the velocity of m2 just before the collision.
Step 3: Applying the conservation of energy
According to the conservation of energy principle, the initial energy equals the final energy, since no energy is lost in an elastic collision.
E_initial = E_final
m1 * g * h = 0.5 * m2 * v^2
Step 4: Solving for h
Rearranging the equation and solving for h, we get:
h = (0.5 * m2 * v^2) / (m1 * g)
Step 5: Finding the velocity v
To find the velocity v, we can use the principle of conservation of momentum. The momentum before the collision equals the momentum after the collision.
Initial momentum = Final momentum
m1 * v1 = m1 * v1' + m2 * v2'
where v1 is the initial velocity of m1, v1' is the final velocity of m1, and v2' is the final velocity of m2.
Since m1 and m2 collide elastically and have equal masses, v1' = -v1 and v2' = 0.
This gives us:
m1 * v1 = -m1 * v1
v1 = -v1
Step 6: Substituting values and calculating
Given that m1 = 5.00 kg and m2 = 10.0 kg, we substitute the values into the equations to find the maximum height h.
v1 = -v1 = -√[(2 * m2 * g) / (m1 + m2)]
v2 = 0
h = (0.5 * m2 * v^2) / (m1 * g)
Substituting v1 and v2 into the equation for h, we get:
h = (0.5 * m2 * (-v1)^2) / (m1 * g)
Now, we substitute the known values:
h = (0.5 * 10.0 kg * (√[(2 * 10.0 kg * 9.8 m/s^2) / (5.00 kg + 10.0 kg)])^2) / (5.00 kg * 9.8 m/s^2)
Calculating this expression will give us the maximum height to which m1 rises after the elastic collision.
To calculate the maximum height to which m1 rises after the elastic collision, we need to analyze the conservation of mechanical energy.
1. Determine the initial and final mechanical energies:
- Initial mechanical energy (Ei): This includes the kinetic energy of both blocks before the collision and the potential energy due to gravity.
- Final mechanical energy (Ef): This includes the potential energy of m1 at its maximum height after the collision.
2. Calculate the initial mechanical energy (Ei):
- The initial kinetic energy of m1 is given by the formula: KE1 = (1/2) * m1 * v1^2, where v1 is the initial velocity of m1.
- The initial kinetic energy of m2 is zero since it is initially at rest.
- The initial potential energy of m1 is given by the formula: PE1 = m1 * g * h1, where g is the acceleration due to gravity and h1 is the initial height of m1.
3. Determine the final velocity of m1 after the collision:
- Since the collision is elastic, both momentum and kinetic energy are conserved.
- Initially, the magnets repel each other, so m1 moves to the right and m2 moves to the left.
- Let v1' be the final velocity of m1 and v2' be the final velocity of m2.
- By conservation of momentum: m1 * v1 = m1 * v1' + m2 * v2' (since the masses are equal, m1 * v1 = m2 * v2').
- By conservation of kinetic energy: (1/2) * m1 * v1^2 = (1/2) * m1 * v1'^2 + (1/2) * m2 * v2'^2 (since v1 = v2 initially and v1' = v2' finally).
- Solve these equations simultaneously to find v1'.
4. Determine the final mechanical energy (Ef):
- The final potential energy of m1 is given by the formula: PE2 = m1 * g * h2, where h2 is the maximum height to which m1 rises.
- Since the collision is elastic, there is no loss of mechanical energy during the collision. Therefore, Ei = Ef.
- Substitute the known values and solve for h2.
5. Calculate the maximum height (h2) to which m1 rises.
Note: To fully solve this problem, you will need the values of mass m1, mass m2, initial height h1, and acceleration due to gravity g.