In a vacuum, a five-rupee coin, a feather of a sparrow bird and a mango are dropped simultaneously from the same height. The time taken by them to reach the bottom it `t_(1), t_(2) and t_(3)` respectively. In this situation, we will observe that

Options

A. `t_(1)gt t_(2)gt t_(3)`
B. `t_(1) gt t_(3) gt t_(3)`
C. `t_(3)gt t_(1)gt t_(2)`
D. `t_(1)=t_(2)=t_(3)`

Explanation

To solve the problem, we need to analyze the situation where a five-rupee coin, a feather, and a mango are dropped simultaneously in a vacuum.

### Step-by-Step Solution:

1. **Understanding the Concept of Vacuum**:
- In a vacuum, there is no air resistance. This means that all objects, regardless of their mass or shape, will fall at the same rate due to gravity.

2. **Identifying the Forces Acting on the Objects**:
- In a vacuum, the only force acting on the objects is gravity. There are no opposing forces like air resistance that would affect their motion.

3. **Applying the Principle of Free Fall**:
- According to the principle of free fall, all objects in a vacuum will accelerate towards the ground at the same rate (approximately 9.81 m/s² on Earth), regardless of their mass.

4. **Time Taken to Reach the Ground**:
- Since all three objects (the five-rupee coin, feather, and mango) are subject to the same gravitational acceleration and there is no air resistance, they will all take the same amount of time to reach the ground.

5. **Conclusion**:
- Therefore, the time taken by the five-rupee coin (t₁), the feather (t₂), and the mango (t₃) will be equal. We can express this mathematically as:
\[ t_1 = t_2 = t_3 \]

### Final Answer:
In this situation, we will observe that \( t_1 = t_2 = t_3 \).

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