A ball of mass 0.35 kg is fired with velocity 157 m/s into the barrel of a spring gun of mass 1.8 kg initially at rest on a frictionless surface. The ball sticks in the barrel at the point of maximum compression of the spring. No energy is lost to friction. What fraction of the ball's initial kinetic energy is stored in the spring?
To find the fraction of the ball's initial kinetic energy stored in the spring, we need to first calculate the initial kinetic energy (KE_initial) of the ball and then determine the final potential energy (PE_spring) stored in the spring.
To calculate the initial kinetic energy (KE_initial), we can use the formula:
KE_initial = (1/2) * mass * velocity^2
where
mass = 0.35 kg (mass of the ball)
velocity = 157 m/s (velocity of the ball)
Plugging in the values into the formula:
KE_initial = (1/2) * 0.35 kg * (157 m/s)^2
Calculating this, we get:
KE_initial = 1734.375 J (rounded to 3 decimal places)
Now, to find the final potential energy (PE_spring) stored in the spring, we need to consider the conservation of momentum.
According to the principle of conservation of momentum, the total initial momentum of the ball and the spring gun is equal to the total final momentum of the ball and the spring gun.
Initial momentum = Final momentum
The initial momentum of the system is given by:
Initial momentum = initial velocity of the ball * mass of the ball + initial velocity of the spring gun * mass of the spring gun
Since the ball is fired into the barrel from rest, the initial velocity of the spring gun is 0 m/s. The equation simplifies to:
Initial momentum = initial velocity of the ball * mass of the ball
The final momentum of the system is given by:
Final momentum = final velocity of the ball + final velocity of the spring gun
Since the ball sticks in the barrel at the point of maximum compression of the spring, the final velocity of the ball and the spring gun is zero. Thus, the equation simplifies to:
Final momentum = 0
Setting the initial momentum equal to the final momentum:
initial velocity of the ball * mass of the ball = 0
Simplifying the equation:
initial velocity of the ball = 0
This tells us that the ball comes to rest immediately after being fired into the barrel.
Since the ball comes to rest, the entire initial kinetic energy is now stored as potential energy in the compressed spring.
Therefore, the fraction of the ball's initial kinetic energy stored in the spring is:
Fraction = PE_spring / KE_initial
Fraction = KE_initial / KE_initial
Fraction = 1
Hence, the fraction of the ball's initial kinetic energy stored in the spring is 1, or 100%.
Use the law of conservation of momentum to compute the final velocity of the gun and ball, Vf.
0.35*157 = (1.8 + 0.35)*Vf
Vf = 25.56 m/s
The final kinetic energy is (25.56/157)^2 = 2.6% of the initial KE
The rest, 97.4%, is stored in the spring, if there was no friction.