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).
Correct Answer: 1.25 - 1.3
Solution and Explanation
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
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