A bomb is dropped from a plane flying horizontally with uniform speed. Show that the bomb will explode vertically below the plane.Is the statement true if the plane flies with uniform speed but not horizontally?

Explanation

To solve the problem, we need to analyze the motion of the bomb and the plane in two scenarios: when the plane is flying horizontally and when it is flying at an angle.

### Step-by-Step Solution:

1. **Understanding the Initial Conditions**:
- The plane is flying horizontally with a uniform speed \( u \).
- At the moment the bomb is dropped, it has the same horizontal velocity \( u \) as the plane.

2. **Analyzing the Bomb's Motion**:
- When the bomb is dropped, it continues to move horizontally with the same velocity \( u \) because there are no horizontal forces acting on it (assuming air resistance is negligible).
- The bomb also starts to fall vertically due to gravity, which accelerates it downward at \( g \).

3. **Time of Flight Calculation**:
- Let \( h \) be the height from which the bomb is dropped.
- The time \( t \) it takes for the bomb to hit the ground can be calculated using the formula for free fall:
\[ h = \frac{1}{2} g t^2 \]
- Rearranging gives:
\[ t = \sqrt{\frac{2h}{g}} \]

4. **Horizontal Distance Travelled**:
- During the time \( t \), the horizontal distance travelled by both the bomb and the plane can be calculated as:
\[ \text{Distance} = u \cdot t \]
- Since both the bomb and the plane have the same horizontal velocity \( u \), they will cover the same horizontal distance in time \( t \).

5. **Conclusion for Horizontal Flight**:
- Therefore, when the bomb reaches the ground, it will land directly below the point where it was dropped from the plane, confirming that the bomb will explode vertically below the plane.

6. **Considering Non-Horizontal Flight**:
- If the plane is flying at an angle \( \theta \) with respect to the horizontal, the horizontal component of the plane's velocity is \( u \cos \theta \).
- At the moment of dropping, the bomb also has the same horizontal velocity component \( u \cos \theta \).
- The time of flight remains the same, and thus the horizontal distance travelled by both the bomb and the plane will still be equal.

7. **Final Conclusion**:
- In both cases (horizontal flight and angled flight), the bomb will always land directly below the plane when it is dropped. Hence, the statement is true irrespective of whether the plane is flying horizontally or at an angle.