Two bodies of masses `m _(1) and m _(2)` fall from heights `h _(1) and h _(2)` respectively. The ratio of their velocities when the hit the ground is

Options

A. `(h _(1))( h _(2))`
B. ` sqrt ( h _(1))/sqrt( h _(2))`
C. `(m _(1) h _(1))/(m _(2) h _(2))`
D. ` (h _(1) ^(2) )/( h _(2) ^(2))`

Answer

To find the ratio of the velocities of two bodies falling from different heights, we can use the equations of motion under gravity. Here’s a step-by-step solution:

### Step 1: Understand the problem
We have two bodies with masses \( m_1 \) and \( m_2 \) falling from heights \( h_1 \) and \( h_2 \) respectively. We need to find the ratio of their velocities when they hit the ground.

### Step 2: Use the equation of motion
For an object falling freely under gravity, the final velocity \( V \) can be calculated using the equation:
\[ V^2 = U^2 + 2gh \]
where:
- \( V \) = final velocity
- \( U \) = initial velocity (which is 0 for free fall)
- \( g \) = acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \))
- \( h \) = height from which the object is falling

Since the initial velocity \( U = 0 \), the equation simplifies to:
\[ V^2 = 2gh \]

### Step 3: Calculate the final velocities
For the first body falling from height \( h_1 \):
\[ V_1^2 = 2gh_1 \]
For the second body falling from height \( h_2 \):
\[ V_2^2 = 2gh_2 \]

### Step 4: Find the ratio of the velocities
To find the ratio of the velocities \( V_1 \) and \( V_2 \), we take the square root of the ratio of their squares:
\[ \frac{V_1^2}{V_2^2} = \frac{2gh_1}{2gh_2} = \frac{h_1}{h_2} \]
Thus, we have:
\[ \frac{V_1}{V_2} = \sqrt{\frac{h_1}{h_2}} \]

### Step 5: Final result
The ratio of the velocities of the two bodies when they hit the ground is:
\[ \frac{V_1}{V_2} = \sqrt{\frac{h_1}{h_2}} \]