A cue stick has a mass of 0.5 kg. The cue stick hits a ball with a mass of 0.2 kg at a velocity of 2.5 m/s. What is the velocity of the ball after it is hit? (1 point) Responses 8.3 m/s 8.3 m/s 3.6 m/s 3.6 m/s 6.3 m/s 6.3 m/s 2.5 m/s 2.5 m/s
To solve this problem, we can use the conservation of momentum principle, which states that the total momentum before an event is equal to the total momentum after the event.
The momentum of an object is given by the product of its mass and velocity.
Let's denote the velocity of the ball after it is hit as v.
The initial momentum of the cue stick and the ball is given by:
Initial momentum = (mass of cue stick) × (velocity of cue stick) + (mass of ball) × (velocity of ball before being hit)
The final momentum of the cue stick and the ball is given by:
Final momentum = (mass of cue stick) × (final velocity of cue stick) + (mass of ball) × (final velocity of ball)
Since the total momentum before and after the event should be equal, we can set up the following equation:
(mass of cue stick) × (velocity of cue stick) + (mass of ball) × (velocity of ball before being hit) = (mass of cue stick) × (final velocity of cue stick) + (mass of ball) × (final velocity of ball)
Plugging in the given values:
(0.5 kg) × (2.5 m/s) + (0.2 kg) × (2.5 m/s) = (0.5 kg) × (final velocity of cue stick) + (0.2 kg) × (final velocity of ball)
Simplifying the equation:
1.25 kg·m/s + 0.5 kg·m/s = (0.5 kg) × (final velocity of cue stick) + (0.2 kg) × (final velocity of ball)
1.75 kg·m/s = (0.5 kg) × (final velocity of cue stick) + (0.2 kg) × (final velocity of ball)
Since the cue stick and the ball have only one dimension of motion, their final velocities in that dimension must have the same magnitude but opposite signs. Therefore, we can write:
(final velocity of cue stick) = -(final velocity of ball)
Substituting this into the equation:
1.75 kg·m/s = (0.5 kg) × (-(final velocity of ball)) + (0.2 kg) × (final velocity of ball)
1.75 kg·m/s = -(0.5 kg + 0.2 kg) × (final velocity of ball)
1.75 kg·m/s = -(0.7 kg) × (final velocity of ball)
1.75 kg·m/s = -0.7 kg·m/s × (final velocity of ball)
Dividing both sides of the equation by -0.7 kg·m/s:
1.75 kg·m/s / -0.7 kg·m/s = final velocity of ball
-2.5 m/s = final velocity of ball
Therefore, the velocity of the ball after it is hit is -2.5 m/s.