A small block of mass m1 = 0.400 kg is released from rest at the top of a curved-shaped frictionless wedge of mass m2 = 3.00 kg, which sits on a frictionless horizontal surface as in Figure P9.60a. When the block leaves the wedge, its velocity is measured to be v1 = 5.00 m/s to the right, as in Figure P9.60b.
Figure P9.60
(a) What is the velocity of the wedge after the block reaches the horizontal surface?
answer should be in m/s to the left
(b) What is the height h of the wedge?
answer should be in m
To solve this problem, we can use the principle of conservation of momentum and conservation of energy.
Let's first consider part (a) to find the velocity of the wedge after the block reaches the horizontal surface.
1. Conservation of momentum:
The initial momentum of the block and the wedge system is zero since they are both at rest initially. When the block leaves the wedge, it has a velocity of 5.00 m/s to the right. To find the velocity of the wedge, we can use the principle of conservation of momentum.
Since momentum is conserved in the absence of external forces, the total momentum before and after the block leaves the wedge should still be zero.
Let's denote the velocity of the wedge as V. The momentum of the block after leaving the wedge is m1 * v1 (mass of the block multiplied by its velocity). The momentum of the wedge after the block leaves is m2 * V (mass of the wedge multiplied by its velocity).
So, we have:
m1 * v1 + m2 * V = 0
Substituting the given values:
0.400 kg * 5.00 m/s + 3.00 kg * V = 0
Now, solve for V (the velocity of the wedge):
2.00 kg m/s + 3.00 kg * V = 0
3.00 kg * V = -2.00 kg m/s
V = -2.00 kg m/s / 3.00 kg
Therefore, the velocity of the wedge after the block reaches the horizontal surface is approximately -0.667 m/s to the left.
Moving on to part (b), let's find the height h of the wedge.
2. Conservation of energy:
When the block slides down the curved-shaped wedge, the potential energy is converted into kinetic energy.
The initial potential energy of the block at the top of the wedge is m1 * g * h (mass of the block multiplied by acceleration due to gravity multiplied by the height).
The final kinetic energy of the block when it leaves the wedge is (1/2) * m1 * v1^2 (one-half multiplied by mass of the block multiplied by the square of its velocity).
Since there is no friction or other energy loss, the total initial potential energy should be equal to the final kinetic energy. Mathematically, we have:
m1 * g * h = (1/2) * m1 * v1^2
Substituting the given values:
0.400 kg * 9.81 m/s^2 * h = (1/2) * 0.400 kg * (5.00 m/s)^2
Simplifying:
3.924 h = 5.00^2 / 2
3.924 h = 6.250
h = 6.250 / 3.924
Therefore, the height h of the wedge is approximately 1.59 meters.
To solve the problem, we can use the principle of conservation of linear momentum. In this case, the total momentum before and after the block leaves the wedge should be equal.
Let's break down the solution step-by-step:
Step 1: Find the initial momentum before the block leaves the wedge.
The initial momentum of the system is given by the product of the mass of the block and its initial velocity:
Initial momentum = m1 * 0 = 0 kg*m/s
Step 2: Find the final momentum after the block leaves the wedge.
The final momentum of the system is given by the product of the mass of the block and its final velocity:
Final momentum = m1 * v1
Step 3: Find the velocity of the wedge after the block reaches the horizontal surface.
According to the conservation of linear momentum, the total initial momentum should be equal to the total final momentum.
0 = m1 * v1 + m2 * v2
Rearranging the equation, we can solve for v2:
v2 = -(m1 * v1) / m2
Step 4: Calculate the velocity of the wedge.
Substitute the given values into the equation:
v2 = -(0.4 kg * 5 m/s) / 3 kg
v2 = -0.6667 m/s
Therefore, the velocity of the wedge after the block reaches the horizontal surface is 0.6667 m/s to the left.
Step 5: Find the height h of the wedge.
We can use the principle of conservation of mechanical energy to find the height of the wedge.
The initial potential energy of the block at the top of the wedge is given by: m1 * g * h, where g is the acceleration due to gravity (9.8 m/s^2).
The final potential energy of the block on the horizontal surface is zero.
Since there is no external work done on the system, the initial potential energy is equal to the final kinetic energy of the system:
m1 * g * h = (1/2) * m1 * v1^2 + (1/2) * m2 * v2^2
Simplify the equation and solve for h:
h = (1/2) * [(v1^2 * m1 + v2^2 * m2) / g]
Substitute the given values into the equation:
h = (1/2) * [(5^2 * 0.4 kg + (-0.6667)^2 * 3 kg) / 9.8 m/s^2]
h ≈ 0.127 m
Therefore, the height h of the wedge is approximately 0.127 m.