How to: toss a ball into the bucket

Classical Mechanics Level 5

A ball rolls off a table with a speed of 2 f t / s 2ft/s . The table is 3.5 3.5 ft high.

Suppose the ball rebounds from the floor at the same angle with which it hits the floor, but loses 20 20 % of its speed due to energy absorbed by the ball on impact. Where does the ball strike the floor on the second bounce?

Enter the answer as the displacement of the ball in feet to the right of the table's edge.

Details and Assumptions :

  • g = 32.1740 f t / s 2 g=32.1740ft/s^2
  • The table is perfectly horizontally flat (and so is the ground) - the picture is not drawn to scale.
  • Round the answer to the nearest hundredth


The answer is 2.13.

John M. by John M. , 24, USA

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2 solutions

Ayush Verma
Oct 28, 2014

Let time of first & second flight are t 1 & t 2 { t }_{ 1 }\quad \& \quad { t }_{ 2 }

1 2 g t 1 2 = h ⇒ t 1 = 2 × 3.5 32.174 = 0.46644 s e c R 1 = u t 1 = 2 × 0.46644 = 0.93288 f t \cfrac { 1 }{ 2 } g{ { t }_{ 1 } }^{ 2 }=h\\ \\ \Rightarrow { t }_{ 1 }=\sqrt { \cfrac { 2\times 3.5 }{ 32.174 } } =0.46644sec\\ \\ { R }_{ 1 }=u{ t }_{ 1 }=2\times 0.46644=0.93288ft

Let speed just before rebound is v.

The path of ball will be same if ball will be thrown towards table from that point with speed v so,

R 1 = 1 2 v 2 s i n 2 θ g ( m u l t i p l i e d b y 1 2 a s o n l y h a l f f l i g h t ) { R }_{ 1 }=\cfrac { 1 }{ 2 } \cfrac { { v }^{ 2 }sin2\theta }{ g } \quad \quad (multiplied\quad by\quad \cfrac { 1 }{ 2 } \quad as\quad only\quad half\quad flight)

As it loses 20% speed,

R 2 = ( 4 5 v ) 2 s i n 2 θ g { R }_{ 2 }=\cfrac { { \left( \cfrac { 4 }{ 5 } v \right) }^{ 2 }sin2\theta }{ g }

R 2 = 16 25 v 2 s i n 2 θ g = 32 25 R 1 ⇒ A n s = R 1 + R 2 = ( 1 + 32 25 ) R 1 = 57 25 × 0.93288 = 2.127 f t { R }_{ 2 }=\cfrac { 16 }{ 25 } \cfrac { { v }^{ 2 }sin2\theta }{ g } =\cfrac { 32 }{ 25 } { R }_{ 1 }\\ \\ \Rightarrow Ans={ R }_{ 1 }+{ R }_{ 2 }=\left( 1+\cfrac { 32 }{ 25 } \right) { R }_{ 1 }=\cfrac { 57 }{ 25 } \times 0.93288=2.127ft

3 Helpful 0 Interesting 0 Brilliant 0 Confused

Nice solution ! I did it the same way. :)

Keshav Tiwari - 6 years, 6 months ago

What I really like about this question is that it is possible to solve it without trigonometry. Great question!

Julian Poon - 6 years, 6 months ago

Did the ball increase it's velocity before hitting the ground?

Mbah Abal - 5 years, 1 month ago

1 Helpful 0 Interesting 0 Brilliant 0 Confused

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