A softball of mass 0.220 kg that is moving with a speed of 4.0 m/s (in the positive direction) collides head-on and elastically with another ball initially at rest. Afterward the incoming softball bounces backward with a speed of 1.8 m/s.
a) Calculate the velocity of the target ball after the collision.
b) Calculate the mass of the target ball.
initial momentum = .22*4 = .88 kg m/s =final
initial Ke =(1/2).22(16) = 1.76 J = final
final momentum = .88=.22(-1.8) + m v
final Ke =1.76 = (1/2).22(1.8^2) + .5 m v^2
solve two eqns for m and v
To solve both parts of this question, we can use the principle of conservation of momentum and the principle of conservation of kinetic energy.
a) Velocity of the target ball after the collision:
According to the principle of conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision. Mathematically, this can be represented as:
(m1 * v1) + (m2 * v2) = (m1 * v1') + (m2 * v2')
where m1 and m2 are the masses of the softball and target ball respectively, v1 and v2 are the initial velocities of the softball and target ball, and v1' and v2' are their final velocities after the collision.
In this case, the initial velocity of the target ball is 0 m/s since it is at rest. The initial velocity of the softball is 4.0 m/s in the positive direction. The final velocity of the softball is -1.8 m/s because it bounces backward.
Substituting these values into the equation, we get:
(0.220 kg * 0 m/s) + (m2 * 0 m/s) = (0.220 kg * -1.8 m/s) + (m2 * v2')
Since the target ball is at rest after the collision, its final velocity v2' is 0 m/s. Plugging this into the equation, we can solve for the mass of the target ball m2:
0 + 0 = (0.220 kg * -1.8 m/s) + (m2 * 0 m/s)
0 = -0.396 kg*m/s
0 = -0.396 kg*m/s
Since the momenta on both sides of the equation are equal, we conclude that m2 * 0 m/s = 0 kg*m/s.
Therefore, the velocity of the target ball after the collision is 0 m/s.
b) Mass of the target ball:
To find the mass of the target ball, we can use the principle of conservation of kinetic energy. According to this principle, the total kinetic energy before the collision is equal to the total kinetic energy after the collision.
The initial kinetic energy is given by:
KE_initial = (1/2) * m1 * v1^2
where m1 is the mass of the softball and v1 is its initial velocity.
The final kinetic energy is given by:
KE_final = (1/2) * m1 * v1'^2 + (1/2) * m2 * v2'^2
where v1' and v2' are the final velocities of the softball and target ball respectively.
Since the velocity of the target ball after the collision is 0 m/s (as calculated in part a), the final kinetic energy is:
KE_final = (1/2) * m1 * v1'^2 + (1/2) * m2 * 0^2
KE_final = (1/2) * m1 * v1'^2
Since the initial kinetic energy and the final kinetic energy are equal, we have:
(1/2) * m1 * v1^2 = (1/2) * m1 * v1'^2
Simplifying the equation:
m1 * v1^2 = m1 * v1'^2
Dividing both sides of the equation by v1^2:
m1 = m1 * (v1'^2 / v1^2)
Plugging in the given values:
m1 = 0.220 kg * (1.8 m/s)^2 / (4.0 m/s)^2
m1 = 0.220 kg * 0.81 / 16
m1 = 0.011025 kg
Therefore, the mass of the target ball is approximately 0.011 kg.