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... 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 ratio of ranges of projectiles is? Read more

Steps to Solve

1. Establish variables for kinetic energies and heights Let ( KE_1 ) and ( KE_2 ) be the kinetic energies of the two particles. Given the ratio ( KE_1 : KE_2 = 2 : 3 ), we can express them as: $$ KE_1 = 2k $$ $$ KE_2 = 3k $$ for some constant ( k ).

2. Relate kinetic energy to the height The maximum height ( h ) of a projectile can be determined using the kinetic energy formula: $$ KE = \frac{1}{2} m v^2 $$ Given that both particles are projected with the same speed, we can write the maximum height for each particle as: $$ h_1 = \frac{KE_1}{mg} \quad \text{and} \quad h_2 = \frac{KE_2}{mg} $$ Substituting the expressions for kinetic energies yields: $$ h_1 = \frac{2k}{mg} \quad \text{and} \quad h_2 = \frac{3k}{mg} $$

3. Substitute the height ratio Given the ratio of maximum heights is ( h_1 : h_2 = 9 : 2 ), we can express this as: $$ \frac{h_1}{h_2} = \frac{\frac{2k}{mg}}{\frac{3k}{mg}} = \frac{2}{3} $$

4. Combine ratios From the height ratio: $$ \frac{h_1}{h_2} = \frac{9}{2} $$ From the kinetic energy ratio: $$ \frac{KE_1}{KE_2} = \frac{2}{3} $$ Now we have two ratios to work with.

5. Calculate the ratio of ranges The range ( R ) of a projectile is given by: $$ R = \frac{v^2}{g} \cdot \sin(2\theta) $$ Since the speed is constant, the ratio of the ranges can be derived from the ratios of heights: $$ \frac{R_1}{R_2} = \sqrt{\frac{h_1}{h_2}} \cdot \left( \frac{KE_1}{KE_2} \right)$$

Substituting in the previous values leads us to: $$ \frac{R_1}{R_2} = \sqrt{\frac{9}{2}} \cdot \frac{2}{3} $$

6. Final calculation Calculating the final ratio: $$ \frac{R_1}{R_2} = \left( \frac{3}{\sqrt{2}} \cdot \frac{2}{3} \right) = \frac{2}{\sqrt{2}} = \sqrt{2} $$