Question

A man throws a ball vertically up in the air with an initial velocity v₁ such that it reaches a height of 12 m with a speed of 12 m/s. If he throws the same ball vertically up with an initial velocity v such that it reaches a maximum height of 12 m. Calculate v₁/v₂ (up to 2 decimal places).

Answer 1

Correct Answer: 1.25 - 1.3

To solve this problem, we use the equations of motion for a vertically thrown object under gravity. Initially, we apply the kinematic equation: v² = u² - 2gh, where v is the final velocity, u is the initial velocity, g is the acceleration due to gravity (approximately 9.81 m/s²), and h is the maximum height.

Given: Initial velocity v₁ makes the ball reach a height h = 12 m with final speed v = 12 m/s. Applying the equation:

12² = v₁² - 2 * 9.81 * 12

144 = v₁² - 235.44

v₁² = 379.44

v₁ = √379.44 ≈ 19.48 m/s

Next, considering the initial velocity v₂ for which the ball reaches exactly 12 m with zero final velocity, we use:

0 = v₂² - 2 * 9.81 * 12

v₂² = 235.44

v₂ = √235.44 ≈ 15.35 m/s

Finally, we calculate the ratio v₁/v₂ as:

v₁/v₂ = 19.48 / 15.35 ≈ 1.27