A softball of mass 0.200 kg that is moving with a speed of 8.2 m/s collides head-on and elastically with another ball initially at rest. Afterward the incoming softball bounces backward with a speed of 3.9 m/s.
1. Calculate the velocity of the target ball after the collision.
2. Calculate the mass of the target ball.
Conservation of momentum leads to
.2kg*8.2m/s=.2kg*(-3.9)m/s + Mass*V
Solve for V in therms of all the other.
Then put that into this for V
conservation of energy.
1/2 .2 *8.2^2=1/2 .2*(3.9)^2 + 1/2 MV^2
then solve for mass, after that, go back and solve for V
Given:
M1 = 0.200kg, V1 = 8.2 m/s.
M2 = ?, V2 = 0.
V3 = -3.9 m/s = Velocity of M1 after the collision. +
V4 = ?, = Velocity of V2 after the collision.
Momentum before = Momentum after.
M1*V1 + M2*V2 = M1*V3 + M2*V4.
0.2*8.2 + M2*0 = 0.2*(-3.9) + M2*V4,
1.64 + 0 = -0.78 + M2*V4,
Eq1: M2*V4 = 2.42.
Conservation of KE Eq:
V4 = (V2(M1-M2) + 2M1*V1)/(M1+M2).
V4 = (0(M1-M2) + 3.28)/(0.2+M2),
V4 = 3.28/(0.2+M2).
In Eq1, replace V4 with 3.28/(0.2+M2) and solve for M2:
M2*(3.28/(0.2+M2) = 2.42.
Multiply both sides by (0.2+M2)
M2*3.28 = 2.42(0.2+M2),
M2*3.28 = 0.484 + M2*2.42,
M2*0.86 = 0.484,
M2 = 0.563kg.
V4 = 3.28/(0.2+0.563) = 4.30 m/s.
KE before and after the collision:
KEb = 0.5*0.2*8.2^2 = 6.72 J.
KEa = 0.5*0.2*(-3.9^2) + 0.5*0.563*4.3^2 = 6.73 J.
To solve this problem, we can use the principles of conservation of momentum and kinetic energy.
1. To calculate the velocity of the target ball after the collision, we can use the principle of conservation of momentum. The total momentum before the collision is equal to the total momentum after the collision.
The initial momentum can be calculated as the product of the mass and velocity of the incoming softball:
Initial momentum = (mass of incoming softball) * (initial velocity of incoming softball)
The final momentum can be calculated as the product of the mass and velocity of the incoming softball after the collision, plus the product of the mass and velocity of the target ball after the collision:
Final momentum = (mass of incoming softball) * (final velocity of incoming softball) + (mass of target ball) * (final velocity of target ball)
Since the collision is elastic, the kinetic energy is conserved. The initial kinetic energy is equal to the final kinetic energy.
The initial kinetic energy can be calculated as:
Initial kinetic energy = 0.5 * (mass of incoming softball) * (initial velocity of incoming softball)^2
The final kinetic energy can be calculated as the sum of the kinetic energy of the incoming softball after the collision and the kinetic energy of the target ball after the collision:
Final kinetic energy = 0.5 * (mass of incoming softball) * (final velocity of incoming softball)^2 + 0.5 * (mass of target ball) * (final velocity of target ball)^2
Using these equations, we can solve for the two unknowns: the final velocity of the target ball and the mass of the target ball.
2. To calculate the mass of the target ball, you can use the equation for final momentum mentioned above and substitute the known values of mass and velocity for the incoming softball and the final velocity for the incoming softball. Solve for the mass of the target ball.