A hunter aims his gun and fires a bullet directly toward's a monkey sitting on a distance tree. At the intant the bullet leaves the barrel of the gun, the monkey drops from the tree :
(i) Will the bullet hit the aim ?
(ii) What Path of bullet will be appeared to monkey ?
(iii) If monkey does not drop from the tree will bullet hit the aim ?

Answer

To solve the problem step-by-step, we will analyze the situation involving the hunter, the bullet, and the monkey.

### Step 1: Understanding the Situation
- A hunter aims his gun at a monkey sitting on a tree.
- The moment the bullet is fired, the monkey drops from the tree.
- We need to determine:
(i) Will the bullet hit the monkey?
(ii) What path will the bullet appear to take from the monkey's perspective?
(iii) If the monkey does not drop, will the bullet hit the aim?

### Step 2: Analyzing the Bullet's Path
- When the bullet is fired, it follows a projectile motion path due to gravity.
- The bullet's vertical position can be described by the equation of motion:
\[ y_b = x \tan(\theta) - \frac{g x^2}{2u^2 \cos^2(\theta)} \]
where:
- \( y_b \) is the vertical position of the bullet,
- \( x \) is the horizontal distance traveled,
- \( \theta \) is the angle of projection,
- \( g \) is the acceleration due to gravity,
- \( u \) is the initial speed of the bullet.

### Step 3: Analyzing the Monkey's Fall
- The monkey drops from the tree and falls freely under gravity.
- The vertical position of the monkey can be described by:
\[ y_m = h - \frac{1}{2} g t^2 \]
where:
- \( h \) is the initial height of the monkey,
- \( t \) is the time taken to fall.

### Step 4: Time of Flight
- The time \( t \) for the bullet to travel horizontally to the monkey can be calculated using:
\[ t = \frac{x}{u \cos(\theta)} \]

### Step 5: Equating the Displacements
- For the bullet to hit the monkey, their vertical displacements must be equal at the same time \( t \):
\[ y_b = y_m \]
- Substituting the expressions for \( y_b \) and \( y_m \) into this equation allows us to check if they will be equal.

### Step 6: Conclusion for Part (i)
- After substituting and simplifying, we find that the vertical displacements of both the bullet and the monkey are equal at the same time. Therefore, **the bullet will hit the monkey**.

### Step 7: Path of the Bullet as Seen by the Monkey (Part ii)
- Since both the bullet and the monkey are falling under the influence of gravity, the relative acceleration between them is zero.
- Therefore, the bullet will appear to the monkey to follow a **straight path**.

### Step 8: Conclusion for Part (iii)
- If the monkey does not drop from the tree, the bullet will follow its projectile path and will not hit the monkey. The bullet will miss the monkey because it would be at a fixed height while the bullet follows a parabolic trajectory.

### Final Answers:
(i) Yes, the bullet will hit the monkey.
(ii) The bullet will appear to follow a straight path to the monkey.
(iii) No, if the monkey does not drop, the bullet will not hit the aim.