Calculate the percentage increase in horizontal range if the maximum height attained by a projectile is inctreased by `20%` by increasing the speed of projectile, assuming that there is no change in angle of projection .

Answer

To solve the problem of calculating the percentage increase in horizontal range when the maximum height attained by a projectile is increased by 20%, we can follow these steps:

### Step 1: Understand the formulas for maximum height and range
The maximum height \( H \) of a projectile is given by the formula:
\[ H = \frac{u^2 \sin^2 \theta}{2g} \]
where \( u \) is the initial speed, \( \theta \) is the angle of projection, and \( g \) is the acceleration due to gravity.

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

### Step 2: Determine the new maximum height
If the maximum height is increased by 20%, the new maximum height \( H' \) can be expressed as:
\[ H' = H + 0.2H = 1.2H \]
Substituting the formula for \( H \):
\[ H' = 1.2 \left(\frac{u^2 \sin^2 \theta}{2g}\right) = \frac{1.2u^2 \sin^2 \theta}{2g} \]

### Step 3: Relate the new maximum height to the new speed
Let the new speed be \( u' \). The new maximum height in terms of \( u' \) is:
\[ H' = \frac{(u')^2 \sin^2 \theta}{2g} \]
Setting the two expressions for \( H' \) equal gives:
\[ \frac{(u')^2 \sin^2 \theta}{2g} = \frac{1.2u^2 \sin^2 \theta}{2g} \]
Cancelling \( \frac{\sin^2 \theta}{2g} \) from both sides:
\[ (u')^2 = 1.2u^2 \]
Taking the square root:
\[ u' = \sqrt{1.2}u = \frac{\sqrt{6}}{5}u \]

### Step 4: Calculate the new range
Now, substituting \( u' \) into the range formula:
\[ R' = \frac{(u')^2 \sin 2\theta}{g} = \frac{(1.2u^2) \sin 2\theta}{g} = 1.2 \left(\frac{u^2 \sin 2\theta}{g}\right) = 1.2R \]

### Step 5: Calculate the percentage increase in range
The percentage increase in range is given by:
\[ \text{Percentage Increase} = \frac{R' - R}{R} \times 100 \]
Substituting \( R' = 1.2R \):
\[ \text{Percentage Increase} = \frac{1.2R - R}{R} \times 100 = \frac{0.2R}{R} \times 100 = 20\% \]

### Conclusion
Thus, the percentage increase in horizontal range when the maximum height is increased by 20% is **20%**.

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