A small block of mass m1 = 0.500 kg is released from rest at the top of a curve-shaped
frictionless wedge of mass m2 = 3.00 kg, which sits on a frictionless horizontal surface as in the figure
below. When the block leaves the wedge, its velocity is measured to be 4.00 m/s to the right, as in figure
below.
(a) What is the velocity of the wedge after the block reaches the horizontal surface?
(b) What is the height h of the wedge?
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To solve this problem, we can use conservation of momentum and conservation of energy principles.
(a) To find the velocity of the wedge after the block reaches the horizontal surface, we need to calculate the momentum of the system before and after the block leaves the wedge.
Before the block leaves the wedge, the system (consisting of the block and the wedge) is at rest, so the initial momentum of the system is zero.
After the block leaves the wedge, the momentum of the block is given by m1 * v, where m1 is the mass of the block and v is its velocity. In this case, the velocity is given as 4.00 m/s to the right. So, the momentum of the block is 0.500 kg * 4.00 m/s = 2.00 kg·m/s.
According to the principle of conservation of momentum, the total momentum of the system before and after the block leaves the wedge must be the same. Therefore, the momentum of the wedge after the block reaches the horizontal surface is also 2.00 kg·m/s to the right.
(b) To find the height h of the wedge, we can use the principle of conservation of energy.
The initial potential energy of the block is m1 * g * h, where g is the acceleration due to gravity and h is the height of the wedge. The gravitational potential energy is converted into kinetic energy as the block slides down the wedge.
At the bottom of the wedge, the potential energy is completely converted into kinetic energy, which is given by (1/2) * m1 * v^2, where v is the velocity of the block when it leaves the wedge (which is given as 4.00 m/s).
Setting the initial potential energy equal to the final kinetic energy, we have:
m1 * g * h = (1/2) * m1 * v^2
Substituting the given values, we have:
(0.500 kg) * (9.81 m/s^2) * h = (1/2) * (0.500 kg) * (4.00 m/s)^2
Simplifying the equation, we can solve for h:
4.905 * h = 4.00^2
h = (16 / 4.905) m
h ≈ 3.264 m
Therefore, the height h of the wedge is approximately 3.264 meters.