A ball is shot out of a spring and into a hollow ballast which then swings upwards 2.5 cm. The ball has a mass of 0.10 kg, the ballast has a mass of 0.50 kg and the spring constant is 500 N/m.
How fast was the ballast-ball system moving immediately after impact?
How fast was the ball moving immediately before impact?
How far was the spring compressed in order to release the ball with this speed?
To calculate the answers to these questions, we can apply the principles of conservation of momentum and conservation of energy.
1. To find the velocity of the ballast-ball system immediately after impact, we can use the conservation of momentum. The momentum before the impact is equal to the momentum after the impact. The initial momentum of the ballast-ball system can be calculated using:
Initial momentum = Mass of the ball * Velocity of the ball + Mass of the ballast * Velocity of the ballast
Since the ballast is initially at rest and there is no external force acting on the system after the impact, the final momentum is simply:
Final momentum = Mass of the ball * Velocity of the ball
Equating the initial and final momentum, we can solve for the velocity of the ballast-ball system immediately after impact.
2. To find the velocity of the ball immediately before impact, we can again use the conservation of momentum. In this case, only the ball is involved in the collision. The initial momentum of the ball is equal to its final momentum before impact. Since there is no external force acting on the ball after the impact, the final momentum is 0 (since the ball comes to rest after impact). Therefore,
Initial momentum = Mass of the ball * Velocity of the ball
Solving for the velocity of the ball immediately before impact will give us the answer.
3. To find how far the spring was compressed in order to release the ball with a specific speed, we can use the principle of conservation of energy. The potential energy stored in the compressed spring is equal to the kinetic energy of the ball when it is released. We can use the formula for potential energy of a spring:
Potential energy = 0.5 * Spring constant * (Compression distance)^2
Since the potential energy will be converted entirely into kinetic energy, we can equate the potential energy to the kinetic energy of the ball:
Kinetic energy = 0.5 * Mass of the ball * (Velocity of the ball)^2
Rearranging the equations and solving for the compression distance will give us the answer.
By plugging in the given values for mass, spring constant, and displacement, we can compute the velocities and compression distance.
To find the speed of the ballast-ball system immediately after impact, we can use the principle of conservation of momentum.
1. First, we find the momentum before impact:
Momentum before impact = momentum of the ballast + momentum of the ball
Momentum before impact = (mass of ballast)(velocity of ballast) + (mass of ball)(velocity of ball)
Given: mass of ballast = 0.50 kg, mass of ball = 0.10 kg
Since the ball and the ballast are initially stationary, the velocity of both is zero.
Momentum before impact = (0.50 kg)(0 m/s) + (0.10 kg)(0 m/s) = 0 kg⋅m/s
2. Next, we find the momentum after impact. The system swings upwards, so the final velocity of the ballast-ball system can be considered as the upward velocity of the center of mass of the system.
According to the law of conservation of momentum, the momentum before impact is equal to the momentum after impact.
Momentum before impact = Momentum after impact
0 kg⋅m/s = (mass of ballast + mass of ball)(final velocity)
Using the given masses,
0 kg⋅m/s = (0.50 kg + 0.10 kg)(final velocity)
Simplifying,
0 kg⋅m/s = (0.60 kg)(final velocity)
Therefore, the final velocity of the ballast-ball system immediately after impact is 0 m/s.
To find the speed of the ball immediately before impact, we can use the principle of conservation of mechanical energy.
3. We can assume there is no energy loss due to external forces and friction. Thus, the total mechanical energy of the system before impact is equal to the total mechanical energy after impact.
Potential energy before impact = Potential energy after impact
(1/2)(k)(x^2) = (1/2)(m)(v^2)
Given: spring constant (k) = 500 N/m, mass of ball (m) = 0.10 kg, maximum displacement of the ball (x) = 2.5 cm = 0.025 m
(1/2)(500 N/m)(0.025 m)^2 = (1/2)(0.10 kg)(v^2)
Simplifying and solving for v,
v^2 = (500 N/m)(0.000625 m)
v^2 = 0.3125 N⋅m
v ≈ 0.559 m/s
Therefore, the ball was moving with a speed of approximately 0.559 m/s immediately before impact.
4. To find how far the spring was compressed, we can use Hooke's law.
According to Hooke's law:
F = kx
Given: spring constant (k) = 500 N/m, force (F) = mass (m) * acceleration (a)
The acceleration of the ball during compression can be found using Newton's second law:
F = ma
a = F/m
Substituting into Hooke's law,
kx = (F/m)(x)
k = F/x
Using the given masses,
(500 N/m)(x) = (0.10 kg)(9.81 m/s^2)(x)
500 Nm = 0.981 N⋅m
Solving for x,
x = 0.001962 m
Therefore, the spring was compressed approximately 0.001962 m in order to release the ball with the given speed.