A ball is projected vertically upwards with a certain initial speed. Another ball of the same mass is projected at an angle of `60^(@)` with the vertical with the same initial speed. At highest points of their journey, the ratio of their potential energies will be

Options

A. `1:1`
B. `2:1`
C. `3:2`
D. `4:1`

Explanation

To solve the problem of finding the ratio of potential energies of two balls projected with the same initial speed, we will follow these steps:

### Step 1: Understand the scenario
We have two balls:
- Ball 1 is projected vertically upwards.
- Ball 2 is projected at an angle of \(60^\circ\) with the vertical.

Both balls have the same mass \(m\) and are projected with the same initial speed \(v\).

### Step 2: Calculate the maximum height of Ball 1
For Ball 1, which is projected vertically upwards, the maximum height \(h_1\) can be calculated using the formula:
\[ h_1 = \frac{v^2}{2g} \]
where \(g\) is the acceleration due to gravity.

### Step 3: Calculate the potential energy of Ball 1 at maximum height
The potential energy \(PE_1\) of Ball 1 at the highest point is given by:
\[ PE_1 = mgh_1 = mg \left(\frac{v^2}{2g}\right) = \frac{mv^2}{2} \]

### Step 4: Calculate the maximum height of Ball 2
For Ball 2, which is projected at an angle of \(60^\circ\) with the vertical, we need to find the vertical component of the initial velocity. The vertical component \(v_y\) is:
\[ v_y = v \cos(60^\circ) = v \cdot \frac{1}{2} = \frac{v}{2} \]
Now, we can calculate the maximum height \(h_2\) for Ball 2:
\[ h_2 = \frac{(v_y)^2}{2g} = \frac{\left(\frac{v}{2}\right)^2}{2g} = \frac{v^2}{4 \cdot 2g} = \frac{v^2}{8g} \]

### Step 5: Calculate the potential energy of Ball 2 at maximum height
The potential energy \(PE_2\) of Ball 2 at the highest point is:
\[ PE_2 = mgh_2 = mg \left(\frac{v^2}{8g}\right) = \frac{mv^2}{8} \]

### Step 6: Calculate the ratio of potential energies
Now, we can find the ratio of the potential energies \(PE_1\) and \(PE_2\):
\[ \text{Ratio} = \frac{PE_1}{PE_2} = \frac{\frac{mv^2}{2}}{\frac{mv^2}{8}} = \frac{8}{2} = 4 \]

### Final Answer
The ratio of the potential energies of the two balls at their highest points is:
\[ \text{Ratio} = 4:1 \]