Two stones of masses `m` and `2 m` are projected vertically upwards so as to reach the same height. The ratio of the kinetic energies of their projection is

Options

A. `2 : 1`
B. `1 : 2`
C. `4 : 1`
D. `1 : 4`

Explanation

To solve the problem, we need to find the ratio of the kinetic energies of two stones projected vertically upwards, given that they reach the same height. Let's denote the masses of the stones as follows:

- Mass of the first stone, \( m_1 = m \)
- Mass of the second stone, \( m_2 = 2m \)

Let \( u_1 \) be the initial velocity of the first stone and \( u_2 \) be the initial velocity of the second stone. Both stones reach the same height \( h \).

### Step 1: Use the equation of motion to relate initial velocity and height.
The equation of motion for an object projected upwards is given by:
\[ v^2 = u^2 - 2gh \]
where \( v \) is the final velocity at the maximum height (which is 0), \( u \) is the initial velocity, \( g \) is the acceleration due to gravity, and \( h \) is the height reached.

For the first stone:
\[ 0 = u_1^2 - 2gh \implies u_1^2 = 2gh \]

For the second stone:
\[ 0 = u_2^2 - 2gh \implies u_2^2 = 2gh \]

### Step 2: Find the ratio of the initial velocities.
Since both stones reach the same height \( h \), we can equate the expressions for \( u_1^2 \) and \( u_2^2 \):
\[ u_1^2 = 2gh \quad \text{and} \quad u_2^2 = 2gh \]

Thus, we can conclude:
\[ u_1^2 = u_2^2 \]
This implies:
\[ u_1 = u_2 \]

### Step 3: Calculate the kinetic energies of both stones.
The kinetic energy (KE) of an object is given by the formula:
\[ KE = \frac{1}{2} m u^2 \]

For the first stone:
\[ KE_1 = \frac{1}{2} m u_1^2 \]

For the second stone:
\[ KE_2 = \frac{1}{2} (2m) u_2^2 = m u_2^2 \]

### Step 4: Substitute the expressions for \( u_1^2 \) and \( u_2^2 \).
Since \( u_1^2 = 2gh \) and \( u_2^2 = 2gh \):
\[ KE_1 = \frac{1}{2} m (2gh) = mgh \]
\[ KE_2 = m (2gh) = 2mgh \]

### Step 5: Find the ratio of the kinetic energies.
Now we can find the ratio of the kinetic energies:
\[ \frac{KE_1}{KE_2} = \frac{mgh}{2mgh} = \frac{1}{2} \]

### Conclusion:
The ratio of the kinetic energies of the two stones at the time of projection is:
\[ \frac{KE_1}{KE_2} = \frac{1}{2} \]