A softball of mass 0.220 kg that is moving with a speed of 6.5 m/s (in the positive direction) collides head-on and elastically with another ball initially at rest. Afterward it is found that the incoming ball has bounced backward with a speed of 4.8 m/s.
Calculate the mass of the target ball when the final velocity of the target ball is 1.7 m/s
Conservation of momentum tells you that
0.22*6.5 = -0.22*4.8 + m v
mv = 2.486 kg m/s (positive direction)
Conservation of kinetic energy tells you that
0.22[(6.5)^2 - (4.8)^2] = (1/2) m v^2
mv^2 = 8.452 kg m^2/s^2
m = (mv)^2/mv^2 = 0.714 kg
Given:
M1 = 0.220kg, V1 = 6.5 m/s.
M2 = ?, V2 = 0.
V3 = -4.8 m/s = velocity of M1 after the collision.
V4 = 1.7m/s = Velocity of M2 after the collision.
M1*V1 + M2*V2 = M1*V3 + M2*V4.
0.22*6.5 + M2*0 = 0.22*(-4.8) + M2*1.7,
1.43 + 0 = -1.06 + M2*1.7,
M2*1.7 = 2.49,
M2 = 1.46kg.
To solve this problem, we can use the principle of conservation of momentum and the principle of conservation of kinetic energy.
Step 1: Calculate the momentum of the incoming ball before the collision.
Momentum is defined as the product of mass and velocity.
Momentum of incoming ball = mass * velocity = (0.220 kg) * (6.5 m/s)
Step 2: Calculate the momentum of the incoming ball after the collision.
Since the collision is elastic, the total momentum before the collision should be equal to the total momentum after the collision.
Momentum of incoming ball after the collision = mass * velocity = (0.220 kg) * (-4.8 m/s) since the ball bounces backward.
Step 3: Calculate the momentum of the target ball after the collision.
Since the target ball was initially at rest, its momentum before the collision is zero.
Momentum of target ball after the collision = mass_target * velocity_target
Step 4: Set up the conservation of momentum equation.
According to the principle of conservation of momentum:
Momentum_before = Momentum_after
(mass_incoming * velocity_incoming) = (mass_incoming * velocity_incoming) + (mass_target * velocity_target)
Step 5: Substitute the known values into the equation.
(0.220 kg) * (6.5 m/s) = (0.220 kg) * (-4.8 m/s) + (mass_target * 1.7 m/s)
Step 6: Solve for the mass_target.
0.220 kg * 6.5 m/s = 0.220 kg * (-4.8 m/s) + mass_target * 1.7 m/s
1.43 kg·m/s = -1.056 kg·m/s + 1.7 m/s * mass_target
1.43 kg·m/s + 1.056 kg·m/s = 1.7 m/s * mass_target
2.486 kg·m/s = 1.7 m/s * mass_target
Step 7: Divide both sides of the equation by 1.7 m/s to solve for the mass_target.
mass_target = 2.486 kg·m/s / 1.7 m/s
mass_target ≈ 1.464 kg
Therefore, the mass of the target ball is approximately 1.464 kg when the final velocity of the target ball is 1.7 m/s.
To find the mass of the target ball, we can use the principle of conservation of momentum. According to this principle, the total momentum before the collision is equal to the total momentum after the collision.
The momentum (p) of an object is given by the formula:
p = m * v
Where:
p is the momentum,
m is the mass of the object, and
v is the velocity of the object.
In this case, we have two objects: the incoming softball (ball A) and the stationary target ball (ball B). Let's denote the mass of ball A as m_A, the mass of ball B as m_B, the initial velocity of ball A as v_A, and the final velocity of ball B as v_B.
Before the collision, the total momentum is the sum of the individual momenta of the balls:
p_total_initial = p_A_initial + p_B_initial
After the collision, the total momentum is the sum of the individual momenta of the balls:
p_total_final = p_A_final + p_B_final
Since the collision is elastic, the total momentum before and after the collision remains the same:
p_total_initial = p_total_final
Now we can set up the equations using the given values:
Before the collision:
p_A_initial = m_A * v_A
p_B_initial = m_B * 0 (since ball B is at rest)
After the collision:
p_A_final = m_A * (-v_A) (since ball A bounces backward)
p_B_final = m_B * v_B
Now, we can equate the initial and final momenta:
m_A * v_A + m_B * 0 = m_A * (-v_A) + m_B * v_B
Simplifying the equation:
m_A * v_A = -m_A * v_A + m_B * v_B
Adding m_A * v_A to both sides:
2 * m_A * v_A = m_B * v_B
Dividing both sides by v_B:
m_B = (2 * m_A * v_A) / v_B
Now we can substitute the given values into the equation and calculate the mass of the target ball (m_B):
m_A = 0.220 kg (mass of the incoming ball)
v_A = -4.8 m/s (final velocity of the incoming ball)
v_B = -1.7 m/s (final velocity of the target ball)
m_B = (2 * 0.220 kg * (-4.8 m/s)) / (-1.7 m/s)
m_B ≈ 2.35 kg
Therefore, the mass of the target ball is approximately 2.35 kg.