Question

A ball is dropped from a height \[h\] As it bounces off the floor, its speed becomes \[80\% \] of what it was just before it hit the floor. The ball then rises to a height equal to
A. \[0.8h\]
B. \[0.75h\]
C. \[0.64h\]
D. \[0.50h\]

Answer 1

Hint: We are asked to find the height to which the ball rises. We can start by noting down the given data. We can then move onto finding the final velocity using one of the equations of motion. Then we move onto finding the final velocity at the given speed using the formula. Then we can find the value of distance using one of the equations of motion.

Formulas used:
The formula used to find the final velocity when the initial velocity is zero is,
\[v = - \sqrt {2gh} \]
The formula used to find the distance is given by,
\[{v^2} = {u^2} + 2ah\]
Where, \[h\] is the height from which the ball is dropped and \[u\] is the initial velocity of the ball.

Complete step by step answer:
Let us start by writing down the given information. The height from which the ball is dropped is given as, \[h\]. Since the ball is dropped and not thrown, the initial velocity of the ball will be \[u = 0\]. The speed becomes \[80\% \] of what it was just before it hit the floor.Now we can move onto finding the final velocity using the formula,
\[v = - \sqrt {2gh} \]

We can multiply this with the percentage value with the velocity and get the final value.That is,
\[v = - 0.8\sqrt {2gh} \]
Now this value can be substituted in the formula, \[{v^2} = {u^2} + 2ah\] and find the value of distance.
\[h' = \dfrac{{{v^2}}}{{2g}} \\
\Rightarrow h' = \dfrac{{{{\left( { - 0.8\sqrt {2gh} } \right)}^2}}}{{2g}}\\
\Rightarrow h' = \dfrac{{0.64 \times 2gh}}{{2g}} \\
\therefore h' = 0.64h\]
In conclusion, the ball then rises to a height of \[0.64h\].

Therefore, the correct answer is option C.

Note: The value of initial velocity is taken as zero because the ball is dropped and not thrown. When the ball is thrown, the arm exerts some force on the ball and this force causes acceleration other than the acceleration due to gravity. This makes the initial velocity zero and the only acceleration with which the body falls is the acceleration due to gravity. This type of motion is called free falling motion.