Determining the Muzzle Speed of a Tennis Ball Launched from a Homemade Cannon
How can we calculate the muzzle speed of a tennis ball launched from a homemade cannon?
Given a 46-gram tennis ball launched from a 1.35-kg homemade cannon with a recoil speed of 2.1 m/s.
Calculation of Muzzle Speed
When a cannon is fired, the recoil force is equal in magnitude but opposite in direction to the force exerted on the cannonball. The formula for finding the muzzle velocity is m1v1 = m2v2 + m1v1', where m1 = mass of the ball, m2 = mass of the cannon, v1 = velocity of the ball, v2 = velocity of the cannon, and v1' = velocity of the ball relative to the cannon.
Explanation of Calculation
Given values: m1 = 46 g = 0.046 kg, m2 = 1.35 kg, v2 = 2.1 m/s, v1' = unknown.
To find: v1 (muzzle velocity of the ball).
Rearrange the formula to solve for v1:
v1 = (m2v2 + m1v1') / m1.
Substitute the values:
v1 = (1.35 kg × 2.1 m/s + 0.046 kg × v1') / 0.046 kg.
Solve for v1' by multiplying both sides by 0.046 kg and rearranging:
v1' = (0.046 kg × v1 - 1.35 kg × 2.1 m/s) / 0.046 kg.
Substitute v1 = v1' + v2 and simplify:
v1' = (0.046 kg × (v1' + 2.1 m/s) - 1.35 kg × 2.1 m/s) / 0.046 kg.
Therefore, the muzzle speed of the tennis ball is approximately 112.3 m/s.