Question

A stone falls freely under gravity. It covers distances $h_1, \, h_2 $ and $h_3$ in the first 5 seconds, the next 5 seconds and the next 5 seconds respectively. The relation between $h_1 , \,h_2 $ and $h_3$ is

• A. $h_1 = 2h_2 = 3h_3$
• B. $h_1 = \frac{h_2}3 = \frac{h_3}5$
• C. $h_2 = 3h_1$ and $3h_2 = 3h_2 $
• D. $h_1 = h_2 = h_3$

Answer 1

The Correct Option is B

To determine the relation between h_1, h_2, and h_3, we can employ the equations of motion under gravity. For an object falling freely under gravity, the distance covered in time t can be calculated using the equation:

s = \frac{1}{2}gt^2

where g is the acceleration due to gravity, which is approximately 9.8 \, \text{m/s}^2.

First, we will calculate the distances h_1, h_2 - h_1, and h_3 - h_2, covered in consecutive 5-second intervals:

1. Distance covered in the first 5 seconds, h_1:
   • Using the formula: h_1 = \frac{1}{2} g (5)^2 = \frac{1}{2} \times 9.8 \times 25 = 122.5 \, \text{m}
2. Distance covered in the next 5 seconds (from 5 to 10 seconds), h_2 - h_1:
   • Total distance covered in 10 seconds: \frac{1}{2} g (10)^2 = \frac{1}{2} \times 9.8 \times 100 = 490 \, \text{m}
   • Thus, h_2 = 490 \, \text{m} and h_2 - h_1 = 490 - 122.5 = 367.5 \, \text{m}
3. Distance covered in the next 5 seconds (from 10 to 15 seconds), h_3 - h_2:
   • Total distance covered in 15 seconds: \frac{1}{2} g (15)^2 = \frac{1}{2} \times 9.8 \times 225 = 1102.5 \, \text{m}
   • Thus, h_3 = 1102.5 \, \text{m} and h_3 - h_2 = 1102.5 - 490 = 612.5 \, \text{m}

From these calculations, the distances for each interval are: h_1 = 122.5 \, \text{m}, h_2 - h_1 = 367.5 \, \text{m}, and h_3 - h_2 = 612.5 \, \text{m}.

The ratio of the distances covered in each interval is: h_1 : (h_2 - h_1) : (h_3 - h_2) = 122.5 : 367.5 : 612.5, which simplifies to the proportionalities h_1 = \frac{h_2}{3} = \frac{h_3}{5}.

Thus, the correct relation is: h_1 = \frac{h_2}3 = \frac{h_3}5.