A boy plays with a ball and he drops it from a height of 1.5 m. Every time the ball hits the ground, it bounces back to attain a height 4/5^th of the previous height. The ball does not bounce further if the previous height is less than 50 cm. What is the number of times the ball hits the ground before the ball stops bouncing?

Correct Option (C)

The initial height from which the ball is dropped is 1.5 m, which is equivalent to 150 cm.

The ball bounces back to 4/5^th of its previous height. The criterion for the ball to cease bouncing is when the height attained in the previous bounce is less than 50 cm.

• After the 1^st impact with the ground, the ball bounces to a height of $\frac{4}{5}\times 150\text{ cm}=\mathbf{\text{120 cm}}$.
• After the 2^nd impact, the ball bounces to a height of $\frac{4}{5}\times 120\text{ cm}=\mathbf{\text{96 cm}}$.
• After the 3^rd impact, the ball bounces to a height of $\frac{4}{5}\times 96\text{ cm}=\mathbf{\text{76.8 cm}}$.
• After the 4^th impact, the ball bounces to a height of $\frac{4}{5}\times 76.8\text{ cm}=\mathbf{\text{61.44 cm}}$.
• After the 5^th impact, the ball bounces to a height of $\frac{4}{5}\times 61.44\text{ cm}=\mathbf{\text{49.152 cm}}$.

Since the height attained after the 5^th impact (49.152 cm) is less than 50 cm, the ball will not bounce further after its 6^th impact with the ground. Consequently, the ball hits the ground a total of 6 times before it stops bouncing.

Incorrect Options:

Options A (4), B (5), and D (7) are incorrect. The sequential calculation of bounce heights demonstrates that the ball continues to bounce until the height achieved after the 5^th impact falls below the 50 cm threshold. This condition is met at 49.152 cm, implying that the ball will undergo a 6^th and final impact before ceasing to bounce. Therefore, selecting 4 or 5 would prematurely conclude the counting of impacts, while 7 would incorrectly suggest an additional bounce beyond the established stopping condition.