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 = 700 N/m) compressed by 8.5 cm. Consider the starting position of the ball to be its location when the spring is compressed. How fast would the 65.2 gram ball be moving at a location which is 31.0 cm higher than the starting position? Assume that the ball slides freely--no rolling.
To calculate the speed of the ball at the specified location, we can use the principle of conservation of mechanical energy.
The mechanical energy of the ball is conserved because there are no external forces doing work on it (assuming no friction or air resistance).
The initial mechanical energy of the ball is given by the potential energy stored in the compressed spring, and when the spring is released, this energy is converted into kinetic energy.
First, let's calculate the potential energy stored in the spring when it is compressed by 8.5 cm. The formula for the potential energy stored in a spring is:
Potential Energy = (1/2) * k * x^2
Where k is the spring constant and x is the displacement of the spring from its equilibrium position.
In this case, k = 700 N/m and x = 8.5 cm = 0.085 m.
So, the potential energy of the ball at the starting position is:
Potential Energy = (1/2) * 700 N/m * (0.085 m)^2
= 2.27 J
Since mechanical energy is conserved, this potential energy will be converted entirely into kinetic energy when the ball reaches the higher position. The formula for kinetic energy is:
Kinetic Energy = (1/2) * mass * velocity^2
Where mass is the mass of the ball and velocity is the speed of the ball.
In this case, mass = 65.2 g = 0.0652 kg.
We can set the potential energy equal to kinetic energy and solve for the velocity:
Potential Energy = Kinetic Energy
2.27 J = (1/2) * 0.0652 kg * velocity^2
Now we can solve for the velocity:
velocity^2 = (2.27 J) / (0.0652 kg)
velocity^2 = 34.85 m^2/s^2
velocity = sqrt(34.85 m^2/s^2)
velocity = 5.9 m/s
So, the ball would be moving at a speed of 5.9 m/s at the location 31.0 cm higher than the starting position.