Two identical metal spheres are released from the top of a tower after t seconds of each other such that they fall along the same vertical line. If air resistance is neglected, then at any instant of time during their fall,

Options

A. the difference in their disptacament temain the same
B. the difference between thelr spaeds remains the same
C. the difference between their heights above ground Is proportional to `t^2`
D. the difference between their displacement Is proportional to t

Answer

To solve the problem of two identical metal spheres released from the top of a tower, we will analyze their motion step by step.

### Step 1: Understanding the Problem
We have two identical metal spheres released from a height \( H \) at different times. The first sphere is released at \( t = 0 \) seconds, and the second sphere is released after \( t \) seconds. We need to analyze their motion under the influence of gravity, neglecting air resistance.

### Step 2: Displacement of the First Sphere
For the first sphere, which is released at \( t = 0 \):
- Initial velocity \( u_1 = 0 \)
- Time of fall \( t' \) (where \( t' \) is the time since it was released)

Using the equation of motion:
\[ s_1 = u_1 t' + \frac{1}{2} g t'^2 \]
Since \( u_1 = 0 \):
\[ s_1 = \frac{1}{2} g t'^2 \]

### Step 3: Displacement of the Second Sphere
For the second sphere, which is released after \( t \) seconds:
- Initial velocity \( u_2 = 0 \)
- Time of fall \( t' - t \) (where \( t' \) is the time since the first sphere was released)

Using the same equation of motion:
\[ s_2 = u_2 (t' - t) + \frac{1}{2} g (t' - t)^2 \]
Since \( u_2 = 0 \):
\[ s_2 = \frac{1}{2} g (t' - t)^2 \]

### Step 4: Difference in Displacement
Now, we calculate the difference in displacement between the two spheres:
\[ \Delta s = s_1 - s_2 = \frac{1}{2} g t'^2 - \frac{1}{2} g (t' - t)^2 \]
Expanding \( (t' - t)^2 \):
\[ \Delta s = \frac{1}{2} g t'^2 - \frac{1}{2} g (t'^2 - 2tt' + t^2) \]
\[ \Delta s = \frac{1}{2} g (2tt' - t^2) \]

### Step 5: Analyzing the Options
1. **Difference in displacement remains the same**: This is not true as we derived \( \Delta s = \frac{1}{2} g (2tt' - t^2) \), which depends on \( t' \).
2. **Difference in speeds remains the same**: The speed of the first sphere is \( v_1 = gt' \) and for the second sphere \( v_2 = g(t' - t) \). The difference \( |v_1 - v_2| = gt \), which is constant.
3. **Difference between heights above the ground is proportional to \( t^2 \)**: This is not true as we derived the difference in heights is not proportional to \( t^2 \).
4. **Difference between displacements is proportional to \( t \)**: This is also not true as it is dependent on \( t' \).

### Conclusion
The correct answer is that the difference in speeds remains the same, which corresponds to option 2.