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 = 684 N/m) compressed by 7.8 cm. Consider the starting position of the ball to be its location when the spring is compressed. How fast would the 59.5 gram ball be moving at a location which is 28.8 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 mechanical energy of a system remains constant when only conservative forces (such as gravity and the spring force) are acting on it.
First, let's determine the potential energy of the compressed spring. The potential energy stored in the spring is given by the formula:
Potential energy (U) = (1/2) * k * x^2,
where k is the spring constant and x is the compression or extension of the spring.
In this case, the compression of the spring is 7.8 cm, which is equal to 0.078 m. The spring constant is given as 684 N/m. Plugging these values into the formula, we get:
Potential energy (U) = (1/2) * 684 N/m * (0.078 m)^2
= 1.92 J
The potential energy stored in the compressed spring is 1.92 Joules.
Next, let's determine the change in potential energy as the ball moves to the higher location. This change in potential energy is equal to the gain in kinetic energy of the ball.
Change in potential energy = m * g * h,
where m is the mass of the ball, g is the acceleration due to gravity, and h is the vertical displacement of the ball.
In this case, the mass of the ball is 59.5 grams, which is equal to 0.0595 kg. The vertical displacement is given as 28.8 cm, which is equal to 0.288 m. The acceleration due to gravity is approximately 9.8 m/s^2. Plugging these values into the formula, we get:
Change in potential energy = 0.0595 kg * 9.8 m/s^2 * 0.288 m
= 0.163 J
The change in potential energy as the ball moves to the higher location is 0.163 Joules.
Since mechanical energy is conserved, the initial potential energy (1.92 J) is equal to the final potential energy (0.163 J) plus the final kinetic energy (Kf) of the ball.
1.92 J = 0.163 J + Kf
Simplifying, we find:
Kf = 1.92 J - 0.163 J
= 1.757 J
The final kinetic energy of the ball is 1.757 Joules.
To find the speed of the ball, we can use the formula for kinetic energy:
Kinetic energy (K) = (1/2) * m * v^2,
where m is the mass of the ball and v is its speed.
Plugging in the values, we have:
1.757 J = (1/2) * 0.0595 kg * v^2
Solving for v, we get:
v^2 = (2 * 1.757 J) / 0.0595 kg
v^2 = 58.932 m^2/s^2
Taking the square root of both sides, we find:
v = √(58.932 m^2/s^2)
v ≈ 7.67 m/s
Therefore, the ball would be moving at a speed of approximately 7.67 meters per second at a location 28.8 cm higher than the starting position in the pinball machine.
To calculate the speed of the ball at a specific location, we need to use the principle of conservation of mechanical energy.
The initial potential energy stored in the compressed spring is converted to the kinetic energy of the ball when it is launched.
1. First, let's calculate the potential energy stored in the compressed spring.
The potential energy in a spring is given by the formula: PE = (1/2)kx^2
Where:
PE is the potential energy
k is the spring constant (684 N/m)
x is the displacement from the equilibrium position (0.078 m)
Plugging in the values:
PE = (1/2)(684 N/m)(0.078 m)^2
PE ≈ 2.004 J
2. Next, let's calculate the kinetic energy of the ball at the starting position.
The kinetic energy is given by the formula: KE = (1/2)mv^2
Where:
KE is the kinetic energy
m is the mass of the ball (59.5 g = 0.0595 kg)
v is the velocity of the ball
Since the ball is at rest at the starting position, the kinetic energy is zero.
KE = 0
3. Now, let's calculate the velocity of the ball at the desired location.
The total mechanical energy at any point along the path is equal to the sum of the potential energy and kinetic energy.
At the starting position, the total mechanical energy is equal to the potential energy:
Total mechanical energy = PE = 2.004 J
At the desired location, the total mechanical energy is equal to the sum of the potential energy and kinetic energy:
Total mechanical energy = PE + KE = 2.004 J
Since the ball is sliding freely, there is no loss of mechanical energy due to friction.
4. Finally, let's calculate the velocity at the desired location.
The total mechanical energy at the desired location is equal to the kinetic energy:
Total mechanical energy = KE = (1/2)mv^2
Rearranging the equation and plugging in the values:
v^2 = (2 × Total mechanical energy) / m
v^2 = (2 × 2.004 J) / 0.0595 kg
v^2 ≈ 67.62 m^2/s^2
Taking the square root of both sides:
v ≈ √(67.62 m^2/s^2)
v ≈ 8.22 m/s
Therefore, the ball would be moving at approximately 8.22 m/s at a location 28.8 cm higher than the starting position.