Use the following information to calculate the speed of a ball in a pinball machine at a specific location. The ball is launched by using a spring (k = 703 N/m) compressed by 9.0 cm. Consider the starting position of the ball to be its location when the spring is compressed. How fast would the 50.3 gram ball be moving at a location which is 25.5 cm higher than the starting position? Assume that the ball slides freely--no rolling.
To calculate the speed of the ball at a specific location, we can use the principle of conservation of mechanical energy. The energy stored in the compressed spring is converted into the kinetic energy of the ball.
First, let's calculate the potential energy stored in the compressed spring. The potential energy (PE) in a spring is given by the formula:
PE = 0.5 * k * x^2
Where:
PE = potential energy
k = spring constant (703 N/m)
x = displacement of the spring (9.0 cm = 0.09 m)
Substituting the values, we get:
PE = 0.5 * 703 N/m * (0.09 m)^2
= 0.5 * 703 N/m * 0.0081 m^2
= 2.8446 J (Joules)
This potential energy is equal to the initial kinetic energy (KE) of the ball when it is released from the compressed position. So:
KE = 2.8446 J
Next, let's calculate the potential energy (PE) at the new location, which is 25.5 cm higher than the starting position. The potential energy at any point can be calculated using the formula:
PE = m * g * h
Where:
m = mass of the ball (50.3 g = 0.0503 kg)
g = acceleration due to gravity (9.8 m/s^2)
h = height difference (25.5 cm = 0.255 m)
Substituting the values, we get:
PE = 0.0503 kg * 9.8 m/s^2 * 0.255 m
= 0.1266195 J
Now, since we assume no energy loss due to friction or air resistance, the initial potential energy (2.8446 J) should be equal to the potential energy at the new location (0.1266195 J) plus the final kinetic energy (KE_final) of the ball.
Therefore:
2.8446 J = 0.1266195 J + KE_final
Rearranging the equation:
KE_final = 2.8446 J - 0.1266195 J
= 2.7179805 J
Now, we can use the equation for kinetic energy to calculate the final speed (v_final):
KE_final = 0.5 * m * v_final^2
Substituting the values, we get:
2.7179805 J = 0.5 * 0.0503 kg * v_final^2
Solving for v_final:
v_final^2 = (2.7179805 J) / (0.5 * 0.0503 kg)
= 107.961786 m^2/s^2
Taking the square root of both sides:
v_final = sqrt(107.961786 m^2/s^2)
= 10.39 m/s
Therefore, the ball would be moving at approximately 10.39 m/s at a location 25.5 cm higher than the starting position.