a ball is dropped from a height of 90m on a floor at each collision with the floor the ball loses tenth of its speed plot the speed time graph between t=0 to 8.2

To analyze the motion of a ball dropped from a height of 90 meters, we need to consider how its speed changes with each bounce. The ball experiences free fall due to gravity until it hits the ground, and upon each collision, it loses a tenth of its speed. Let's break this down step by step to plot the speed-time graph from t=0 to 8.2 seconds.

Understanding the Initial Conditions

When the ball is dropped from a height of 90 meters, it starts with an initial speed of 0 m/s. As it falls, it accelerates due to gravity, which is approximately 9.81 m/s². The time it takes to reach the ground can be calculated using the formula:

• h = 0.5 * g * t²

Here, h is the height (90 m), g is the acceleration due to gravity (9.81 m/s²), and t is the time in seconds. Rearranging the formula to solve for t gives:

• t = √(2h/g)

Substituting the values:

• t = √(2 * 90 / 9.81) ≈ 4.27 seconds

Calculating Speed Before Each Bounce

When the ball hits the ground, it will have reached its maximum speed just before impact. The speed just before hitting the ground can be calculated using:

• v = g * t

Substituting the time we found:

• v = 9.81 * 4.27 ≈ 41.9 m/s

Upon hitting the ground, the ball loses a tenth of its speed, so its new speed after the first bounce is:

• v' = v - 0.1 * v = 0.9 * v

Calculating this gives:

• v' ≈ 0.9 * 41.9 ≈ 37.71 m/s

Subsequent Bounces

For each subsequent bounce, we repeat the process. The time taken to fall again will be slightly less due to the reduced speed. However, for simplicity, we can assume the time taken to fall remains approximately the same for the first few bounces, as the height will be less than 90 m but not drastically different for the first few bounces.

After the first bounce, the speed before the second bounce will be:

• v'' = 0.9 * v' ≈ 0.9 * 37.71 ≈ 33.94 m/s

This process continues, with the speed reducing by 10% each time:

• Third bounce: v''' ≈ 30.55 m/s
• Fourth bounce: v'''' ≈ 27.49 m/s
• Fifth bounce: v''''' ≈ 24.74 m/s

Plotting the Speed-Time Graph

Now, let's summarize the speed after each bounce and the time intervals:

• 0 seconds: 0 m/s (initial drop)
• 4.27 seconds: 41.9 m/s (first impact)
• 4.5 seconds: 37.71 m/s (first bounce)
• 8.0 seconds: 33.94 m/s (second impact)
• 8.2 seconds: 30.55 m/s (third impact)

To plot the graph, you would mark these points on a graph with time on the x-axis and speed on the y-axis. The graph will show a steep increase in speed as the ball falls, followed by a series of peaks and drops as it bounces, illustrating the loss of speed with each collision.

Final Thoughts

This analysis provides a clear understanding of how the ball's speed changes over time due to the effects of gravity and energy loss during bounces. By plotting these points, you can visualize the relationship between time and speed effectively.