A billiard ball of mass m = 0.250 kg hits the cushion of a billiard table at an angle of θ1 = 70.0° at a speed of v1 = 2.50 m/s. It bounces off at an angle of θ2 = 63.0° and a speed of v2 = 2.10 m/s.
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To solve this problem, we can use the principle of conservation of momentum and conservation of kinetic energy.
Step 1: Find the initial momentum before the collision
The momentum is given by the equation:
p1 = m * v1,
where p1 is the initial momentum, m is the mass of the billiard ball, and v1 is the initial velocity.
Given values are:
m = 0.250 kg,
v1 = 2.50 m/s.
Substituting these values into the equation, we have:
p1 = 0.250 kg * 2.50 m/s.
Step 2: Find the final momentum after the collision
Using the same equation, the final momentum is given by:
p2 = m * v2,
where p2 is the final momentum, m is the mass of the billiard ball, and v2 is the final velocity.
Given values are:
m = 0.250 kg,
v2 = 2.10 m/s.
Substituting these values into the equation, we have:
p2 = 0.250 kg * 2.10 m/s.
Step 3: Apply the principle of conservation of momentum
According to the principle of conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision. Therefore, we have:
p1 = p2.
Substituting the values of p1 and p2, we get:
0.250 kg * 2.50 m/s = 0.250 kg * 2.10 m/s.
Simplifying the equation, we find:
0.625 kg⋅m/s = 0.525 kg⋅m/s.
Step 4: Find the initial kinetic energy before the collision
The kinetic energy is given by the equation:
KE1 = (1/2) * m * v1^2,
where KE1 is the initial kinetic energy, m is the mass of the billiard ball, and v1 is the initial velocity.
Given values are:
m = 0.250 kg,
v1 = 2.50 m/s.
Substituting these values into the equation, we have:
KE1 = (1/2) * 0.250 kg * (2.50 m/s)^2.
Step 5: Find the final kinetic energy after the collision
Using the same equation, the final kinetic energy is given by:
KE2 = (1/2) * m * v2^2,
where KE2 is the final kinetic energy, m is the mass of the billiard ball, and v2 is the final velocity.
Given values are:
m = 0.250 kg,
v2 = 2.10 m/s.
Substituting these values into the equation, we have:
KE2 = (1/2) * 0.250 kg * (2.10 m/s)^2.
Step 6: Apply the principle of conservation of kinetic energy
According to the principle of conservation of kinetic energy, the total kinetic energy before the collision is equal to the total kinetic energy after the collision. Therefore, we have:
KE1 = KE2.
Substituting the values of KE1 and KE2, we get:
(1/2) * 0.250 kg * (2.50 m/s)^2 = (1/2) * 0.250 kg * (2.10 m/s)^2.
Simplifying the equation, we find:
0.625 J = 0.55125 J.
Step 7: Calculate the change in momentum
To calculate the change in momentum (Δp), subtract the initial momentum from the final momentum:
Δp = p2 - p1.
Substituting the values of p2 and p1, we get:
Δp = 0.525 kg⋅m/s - 0.625 kg⋅m/s.
Simplifying the equation, we find:
Δp = -0.100 kg⋅m/s.
Step 8: Calculate the direction of the change in momentum
The change in momentum is negative (-0.100 kg⋅m/s) because the ball bounces off in the opposite direction.
Therefore, a billiard ball of mass 0.250 kg hits the cushion of a billiard table at an angle of 70.0° at a speed of 2.50 m/s. It bounces off at an angle of 63.0° and a speed of 2.10 m/s with a change in momentum of -0.100 kg⋅m/s.
To determine the change in momentum and the impulse, we can use the principles of conservation of momentum and the laws of physics.
1. Determine the initial momentum:
The initial momentum of an object is given by the product of its mass and velocity. In this case, the initial momentum is:
p1 = m * v1, where m is the mass of the billiard ball and v1 is its initial velocity.
2. Determine the final momentum:
The final momentum can be calculated using the same formula as above, using the mass of the ball and the final velocity:
p2 = m * v2, where v2 is the final velocity of the billiard ball.
3. Determine the change in momentum:
The change in momentum is simply the difference between the final momentum and the initial momentum:
Δp = p2 - p1
4. Determine the impulse:
Impulse is defined as the change in momentum, and can be calculated using the equation:
Impulse = Δp
5. Solve for the impulse:
Plug in the values for p1 and p2 into the equation for impulse, and solve for the change in momentum:
Impulse = p2 - p1
Now, let's calculate the values:
Given:
m = 0.250 kg (mass of the billiard ball)
v1 = 2.50 m/s (initial velocity)
v2 = 2.10 m/s (final velocity)
Using the equations above, we can calculate the initial and final momentum and the change in momentum as follows:
Initial momentum:
p1 = m * v1
p1 = 0.250 kg * 2.50 m/s
Final momentum:
p2 = m * v2
p2 = 0.250 kg * 2.10 m/s
Change in momentum:
Δp = p2 - p1
Plugging in the values:
Δp = (0.250 kg * 2.10 m/s) - (0.250 kg * 2.50 m/s)
Solving for Δp:
Δp = 0.525 kg·m/s - 0.625 kg·m/s
Δp = -0.100 kg·m/s
Therefore, the change in momentum or the impulse exerted on the ball is -0.100 kg·m/s. The negative sign indicates that the direction of the momentum has changed after the collision.