How to Calculate the Horizontal Distance Traveled by a Tennis Ball Launched from a Spring-Loaded Machine

When a tennis ball is launched from a spring-loaded machine, it follows a projectile motion trajectory. To calculate the horizontal distance that the ball can travel, several steps need to be taken into account.

Step 1: Calculate Initial Kinetic Energy

In this case, the initial kinetic energy (KE) of the tennis ball is equal to the elastic potential energy stored in the compressed spring. The formula for elastic potential energy of a spring is 0.5 * k * x^2, where k is the spring constant and x is the displacement of the spring. Substituting the given values, we find KE = 0.5 * 440 N/m * (0.45 m)^2 = 89.1 J.

Step 2: Determine Time to Hit the Ground

Since the only force acting on the ball during its flight is gravity, we can use the formula h = 0.5gt^2 to calculate the time it takes for the ball to hit the ground. Substituting the known values, we get t = sqrt((2 * 1.2 m) / 9.8 m/s^2) = 0.494 s.

Step 3: Calculate Horizontal Distance

With the initial speed of the ball determined using the conservation of mechanical energy (v0 = sqrt((2 * KE) / m)), we can find the horizontal distance traveled by multiplying the initial speed by the time taken. Thus, the horizontal distance is d = 40.86 m/s * 0.494 s = 20.18 m.

By following these steps, one can accurately determine the horizontal distance covered by a tennis ball launched from a spring-loaded machine.