A tennis ball initially moves with a velocity of 27.0 m/s horizontally to the right. A tennis racket strikes the ball, giving it a velocity of 27.0 m/s to the left. Treat the collision as elastic and the ball and racket as particles of mass 0.0580 kg and 0.272 kg respectively.
Find the intial velocity of the racket.
Please help Im stuck
This is an elastic impact problem.
Impact problems have one available equation, namely conservation of momentum.
m1u1+m2u2=m1v1+m2v2.
u, v represent initial and final velocities respectively, and m1, m2 are masses of bodies involved.
Note that velocities have signs, e.g. positive to the right, negative to the left, etc.
For elastic impacts, there is an additional equation, namely the conservation of energy.
Using the same notations above, we have
(1/2)m1u1²+(1/2)m2u2²=(1/2)m1v1²+(1/2)m2v2²
Since m1, m2, u1, v1 are given, these two equations will help you solve for u2 and v2.
For more explanations and examples, see:
http://www.efm.leeds.ac.uk/CIVE/CIVE1140/section04/elastic_impact.html
or
http://hyperphysics.phy-astr.gsu.edu/hbase/elacol.html
To find the initial velocity of the racket, we need to 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 equation for momentum is:
Momentum = Mass * Velocity
Before the collision:
The momentum of the ball before the collision is given by:
Momentum of the ball before = Mass of the ball * Velocity of the ball before
= (0.0580 kg) * (27.0 m/s to the right)
The momentum of the racket before the collision is initially zero since it is at rest before striking the ball.
After the collision:
The momentum of the ball after the collision is given by:
Momentum of the ball after = Mass of the ball * Velocity of the ball after
= (0.0580 kg) * (27.0 m/s to the left)
The momentum of the racket after the collision is given by:
Momentum of the racket after = Mass of the racket * Velocity of the racket after
= (0.272 kg) * (Velocity of the racket)
Since the collision is elastic, the total momentum before the collision is equal to the total momentum after the collision:
Momentum of the ball before = Momentum of the ball after
(0.0580 kg) * (27.0 m/s to the right) = (0.0580 kg) * (27.0 m/s to the left) + (0.272 kg) * (Velocity of the racket)
Simplifying the equation:
(0.0580 kg) * (27.0 m/s) = (0.0580 kg) * (27.0 m/s) + (0.272 kg) * (Velocity of the racket)
To find the velocity of the racket, we can rearrange the equation and solve for it:
(0.272 kg) * (Velocity of the racket) = (0.0580 kg) * (27.0 m/s) - (0.0580 kg) * (27.0 m/s)
Dividing both sides by (0.272 kg) to isolate the velocity of the racket:
Velocity of the racket = [(0.0580 kg) * (27.0 m/s) - (0.0580 kg) * (27.0 m/s)] / (0.272 kg)
Calculating the velocity of the racket:
Velocity of the racket = 0 m/s
Therefore, the initial velocity of the racket is 0 m/s.