In a projectile the statement are true or false
Max. height `h= (R)/(2)tanalpha`
where the letters have the usual meanings.

Answer

To determine whether the statement "Max. height \( h = \frac{R}{2} \tan \alpha \)" is true or false, let's analyze the formulas for the maximum height and the range of a projectile.

### Step 1: Understand the formulas
1. The formula for the maximum height \( h \) of a projectile launched at an angle \( \alpha \) with initial velocity \( u \) is given by:
\[ h = \frac{u^2 \sin^2 \alpha}{2g} \]
where \( g \) is the acceleration due to gravity.

2. The formula for the range \( R \) of the projectile is given by:
\[ R = \frac{u^2 \sin 2\alpha}{g} \]

### Step 2: Substitute the range into the height formula
We need to check if the statement \( h = \frac{R}{2} \tan \alpha \) holds true.

3. First, express \( R \) in terms of \( u \) and \( \alpha \):
\[ R = \frac{u^2 \sin 2\alpha}{g} \]

4. Now, substitute \( R \) into the statement \( h = \frac{R}{2} \tan \alpha \):
\[ h = \frac{1}{2} \cdot \frac{u^2 \sin 2\alpha}{g} \cdot \tan \alpha \]

### Step 3: Simplify the expression
5. Recall that \( \tan \alpha = \frac{\sin \alpha}{\cos \alpha} \). Therefore, we can rewrite the expression:
\[ h = \frac{u^2 \sin 2\alpha \cdot \sin \alpha}{2g \cos \alpha} \]

6. Now, use the identity \( \sin 2\alpha = 2 \sin \alpha \cos \alpha \):
\[ h = \frac{u^2 (2 \sin \alpha \cos \alpha) \sin \alpha}{2g \cos \alpha} \]

7. Simplifying further:
\[ h = \frac{u^2 \sin^2 \alpha}{g} \]

### Step 4: Compare with the original height formula
8. We already know from the maximum height formula that:
\[ h = \frac{u^2 \sin^2 \alpha}{2g} \]

### Step 5: Conclusion
9. Comparing both expressions for \( h \):
- From the maximum height formula: \( h = \frac{u^2 \sin^2 \alpha}{2g} \)
- From the substituted expression: \( h = \frac{u^2 \sin^2 \alpha}{g} \)

Since these two expressions for \( h \) are not equal, we conclude that the original statement is **false**.

### Final Answer
The statement "Max. height \( h = \frac{R}{2} \tan \alpha \)" is **false**.