Subtracting or Adding Real numbers

Question

A ball is dropped from a height of 25.6 feet. After it hits the ground, it bounces to 12.8 feet and falls back to the ground. Next it bounces to 6.4 feet and falls back to the ground. Then it bounces to 3.2 feet and falls back to the ground...

     a)Find the difference in heights between each consecutive bounce.

     b)If the pattern continues, will the ball ever stop bouncing? Explain

Answer 1

a)  Each new height is 1/2 of the previous height.  (25.6 times 0.5 = 12.8, etc.)

b) The bounces will eventually be infinitesimal, so no bounce can be observed.

Answer 2

The starting number is 25.6.   When it starts to hit the ground, it loses height, so we subtract the height lost.  When it bounces up, we add that amount.

 

numbers with negative signs indicate falling to the ground from its previous height.

 

 

25.6 - 25.6 + 12.8 - 12.8 + 6.4 - 6.4 + 3.2 - 3.2

 

 

a)

 

To find the difference in heights between each consecutive bounce, subtract each value of bounce consecutively.

 

Difference between first bounce = 25.6 - 12.8 = 12.8

Difference between second bounce = 12.8 - 6.4 = 6.4

Difference between third bounce = 6.4 - 3.2 = 3.2

 

 

b)

 

If the pattern continues, the ball will stop bouncing.  If we look at the difference between each bounce consecutively, the difference in heights gets smaller.  The smaller the height, the closer an object is closer to the ground.

 

 

If you like, you can take a ball and let it drop from a distance.  See it bounce and you will notice the heights become less and less, until the ball eventually becomes a grounded object, never to rise again.