Ball Bouncing Challenge

Ball Bouncing

On Saturday, Christina and I took a stroll down to the gym to test the velocity and position of a Champion basketball after bouncing it.  

To further help you visualize our collection of data, here's a video of Christina bouncing the ball:

This is what the video looked like with the data points:

Here's a free body diagram of what the ball looked like in the air:

The reason why there are no other forces on the ball is because only the force of gravity is pulling the ball to the earth.

Here's our v v.s. t graph:

On our v v.s. t graph, we were observing two things: the ball's velocity in the vertical direction and the balls' velocity in the horizontal direction.  The data with the light blue border corresponds with the light purple line on the graph.  This line was a representation of the ball's velocity in the horizontal direction.  It is basically a straight line, only a bit of air resistance was slowing it down.  The data with the pink border corresponds with the green line, which depicts the ball's vertical velocity.  As you can see, when the ball hit the ground, it's velocity was at 0, but then increased in the negative direction as it continued on to another bounce.

Here's our x v.s. t graph:

In this graph, you can see how the ball's vertical movement gave us a parabola, showing that it had an acceleration.  The ball's horizontal line showed that the ball moved forward at constant velocity.

Analysis

The acceleration in the y direction was -9.6 m/s^2 (which would make sense, as the average force of gravity is 10N).  This was the slope of our green line in the v v.s. t graph.The acceleration in the x direction was -0.54 m/s^2.  This was the slope of the purple line in the v v.s. t graph.  

The initial velocity in the horizontal direction was about 6 m/s, and the initial velocity in the vertical direction was about 4.6 m/s (the graph is a little bit off, but our data is correct).

The ball's velocity at the top of it's path in the horizontal direction was constant, as no forces (except for minor air resistance) prevented it from moving forward.  The ball's velocity at the top of it's path in the vertical direction was also 0, and you know this because it had been slowing down until it had reached it's peak, where it stopped momentarily before accelerating to the floor.

The final velocity in the vertical direction was about -5 m/s, and the final velocity in the horizontal direction remained at a constant 6 m/s.

The ball's highest point was 2 meters, as represented on our graph of x v.s. t.  In order to solve for the distance travelled in the horizontal direction, you can use:

Δx= 1/2 a t^2 + vi * t

By plugging in -0.54 as the acceleration, 1.05 as the time, and 6 as the initial velocity, you end up with the Δx being 6.01 meters.

It took the ball 0.6 seconds to reach the peak of it's bounce.  The ball was in the air for 1.1 seconds, which is shown on out x v.s. t graph.  

Conclusions

The experiment was pretty impressive, as we studied the balls' motion not only in the forward direction, but in the upward direction as well.  The vertical acceleration was almost equal to the force of gravity, which is 9.8N (or 10N).  The horizontal acceleration however was irrelevant, because the ball bounced at a constant velocity of 6 m/s across the gym.  This justifies that the ball's upwards velocity was not constant, but it's forwards velocity was.  

We solved for the horizontal displacement by using the change in x formula.  By plugging in the acceleration, time, and initial velocity, we were able to conclude that the total horizontal displacement was 6.01 meters.  The ball's velocity at the top of it's path in the vertical direction was 0m/s, as it was changing from going up to going down.  It had been going in the upwards direction, slowing down in the process, until it vertically stops momentarily, and then continues accelerating in the downwards direction.  It's horizontal velocity however remained constant at 6 m/s, as the ball continued moving forward while it's vertical path had acceleration.  To solve for the vertical component, you would also use the 'change in x' formula, and plug in the total time.  However, to solve for the ball's height, you would use the 'change in x' formula, and plug in the amount of time it took to reach it's height.