Calculating the Height at Which a Ball's Speed Decreases to One-Half of Its Initial Value
When a ball is thrown straight upward, its speed changes as it rises, ultimately coming to a maximum height before falling back down. This problem considers the height at which the speed of the ball decreases to one-half of its initial value. By utilizing principles of energy conservation and kinematics, we can determine this specific height. Let's explore the step-by-step solution and the mathematical framework behind it.
Problem Overview
Given:
The maximum height reached by the ball, (h_{text{max}} 12 text{m}). The ball's speed decreases to one-half of its initial value at the height we are looking for.Principles Involved
The problem combines two fundamental principles:
Energy Conservation: The total mechanical energy (kinetic potential) of the ball remains constant in the absence of non-conservative forces (like air resistance). Kinematics: The study of motion without considering the forces that cause it. We will use kinematic equations to solve for the height.Step-by-Step Solution
Step 1: Determine the Initial Velocity (v_0)
At the maximum height, the final velocity (v) is zero. Using the kinematic equation at the peak, we have:
0 (v_0^2 - 2gh_{text{max}})
Solving for (v_0):
(v_0^2 2gh_{text{max}} 2 times 9.81 times 12)
(v_0^2 235.44)
(v_0 sqrt{235.44} approx 15.34 text{m/s})
Step 2: Find the Height at Which the Speed Decreases to One-Half of (v_0)
Let's denote the height at which the speed decreases to half of its initial value as (s). The final velocity at this height is (v frac{1}{2}v_0 approx 7.67 text{m/s}). Using the kinematic equation:
(v^2 v_0^2 - 2gs)
Substituting the values:
(7.67^2) 15.34^2 - 2 times 9.81 times s
((7.67^2)) and ((15.34^2)):
58.7689 235.44 - 2 times 9.81 times s
Rearranging to solve for (s):
2 times 9.81 times s 235.44 - 58.7689
2 times 9.81 times s 176.6711
(s frac{176.6711}{2 times 9.81} approx 9.00 text{m})
Conclusion
The height above the launch point at which the speed of the ball decreases to one-half of its initial value is approximately 9 meters. This solution demonstrates the application of both energy conservation and kinematic principles to determine the height at which a specific velocity condition is met.
Additional Insights
By examining the relationship between the initial and final velocity, and the height, this problem also highlights the importance of understanding motion in one-dimensional space and the role of acceleration due to gravity. The solution can be further extended to explore more complex scenarios, such as including air resistance or considering the ball's deformation.