Two particles of equal mass projected with same speed follow parabolic path near earth surface. The ratio of minimum kinetic energies is 2:3 and ratio of maximum heights is 9:2. Th... Two particles of equal mass projected with same speed follow parabolic path near earth surface. The ratio of minimum kinetic energies is 2:3 and ratio of maximum heights is 9:2. Then the ratio of ranges of projectiles is. Read more

Steps to Solve

1. Understanding the ratios of energies and heights

Given the ratio of minimum kinetic energies, $KE_1:KE_2 = 2:3$, and the ratio of maximum heights, $H_1:H_2 = 9:2$.

2. Use the relationship between kinetic energy and range

The range (R) of a projectile is given by the formula: $$ R = \frac{v^2 \sin(2\theta)}{g} $$ where (v) is the initial velocity, (\theta) is the angle of projection, and (g) is the acceleration due to gravity.

3. Connect kinetic energy to the ratios

The minimum kinetic energy can be expressed as: $$ KE = \frac{1}{2} mv^2 $$ Given that masses are equal and the speed is constant, calculate the ratios as follows: $$ \frac{KE_1}{KE_2} = \frac{2}{3} \implies \frac{v_1^2}{v_2^2} = \frac{2}{3} $$

4. Express speeds in relation to the kinetic energy ratio

Let ( v_2^2 = 3k ) and ( v_1^2 = 2k ), where ( k ) is a constant.

5. Use the height relationship

The height ( H ) can be expressed as: $$ H = \frac{v^2 \sin^2(\theta)}{2g} $$ Using the height ratios given: $$ \frac{H_1}{H_2} = \frac{9}{2} \implies \frac{v_1^2 \sin^2(\theta_1)}{v_2^2 \sin^2(\theta_2)} = \frac{9}{2} $$

6. Combine the equations

From the two ratios:

• Kinetic energy ratio gives us a relationship for speed.
• The height ratio gives us another relationship relating speed and angle.

7. Express ranges in terms of energy and height ratios

We can find the ratio of ranges as: $$ \frac{R_1}{R_2} = \frac{KE_1}{KE_2} \cdot \frac{H_1}{H_2} $$

Substituting the respective ratios: $$ \frac{R_1}{R_2} = \frac{2/3 \cdot 9/2} $$

8. Calculate the ratio of ranges

Now calculate the product: $$ \frac{R_1}{R_2} = \frac{2 \cdot 9}{3 \cdot 2} = \frac{9}{3} = 3 $$

Thus, upon simplification, we find:

• The ratio of ranges is ( \sqrt{3} ).