How Does Physics Explain a Ball Shot from a Spring into a Ballast?

[twjtiger]

This question is blowing my mind, can anyone help?

A ball is shot out of a projector by 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.

a. How fast is the ballast-ball system moving immediately after impact?

b. How fast was the ball moving immediately before impact?

c. How far was the spring compressed in order to release the ball with this speed?

 

[neutrino]

Use conservation of energy for (a) and (c), and linear momentum conservation for (b).

 

[quicknote]

I can help break down this problem for you. First, let's define some variables. The initial velocity of the ball before it is shot out of the projector is 0 m/s. The final velocity of the ballast-ball system after impact can be calculated using conservation of momentum. The equation for this is m1v1 + m2v2 = (m1 + m2)v, where m1 and v1 are the mass and velocity of the ball, m2 and v2 are the mass and velocity of the ballast, and v is the final velocity of the system. Plugging in the values given, we get (0.10 kg)(0 m/s) + (0.50 kg)(v2) = (0.10 kg + 0.50 kg)(v). Solving for v, we get v = 0.1 m/s. This is the speed of the ballast-ball system immediately after impact.

To find the initial velocity of the ball, we can use the conservation of energy principle. The energy before impact (in the form of potential energy from the compressed spring) is equal to the energy after impact (in the form of kinetic energy of the ball). The equation for this is 1/2kx^2 = 1/2mv^2, where k is the spring constant, x is the distance the spring is compressed, m is the mass of the ball, and v is the velocity of the ball. Plugging in the values given, we get (1/2)(500 N/m)(x^2) = (1/2)(0.10 kg)(v^2). Solving for v, we get v = 10 m/s. This is the speed of the ball immediately before impact.

To find the distance the spring was compressed, we can rearrange the equation for conservation of energy to solve for x. This gives us x = √(2m/k)(v^2), where m is the mass of the ball and v is the velocity of the ball. Plugging in the values given, we get x = √(2(0.10 kg)/(500 N/m))(10 m/s)^2 = 0.2 m. This is the distance the spring was compressed in order to release the ball with a speed of 10 m/s.

I hope this helps to clarify the problem and provide a scientific explanation for the