Question

If two stones are thrown vertically upwards with their velocities in the ratio 2:5, then the ratio of the maximum heights attained by the stones is

• A. 4:25
• B. 1:5/2
• C. 6:15
• D. 8:20

Hint:

For projectile motion problems, remember the key relationships:
Maximum height \(H \propto u^2\) (for a given angle)
Time of flight \(T \propto u\)
Range \(R \propto u^2\)
These proportionality relations allow you to solve ratio problems very quickly without calculating the actual values.

Answer 1

The Correct Option is A

Step 1: Conceptual Foundation:
An object projected vertically upward converts initial kinetic energy to gravitational potential energy. At its apex, its velocity is zero. This scenario is analyzed using kinematic equations that relate initial velocity, final velocity, acceleration, and displacement.
Step 2: Governing Equation:
The applicable kinematic equation is:\[ v^2 = u^2 + 2as \]At the maximum altitude (\(H\)), the final velocity (\(v\)) is 0, acceleration (\(a\)) is \(-g\), and displacement (\(s\)) is \(H\).\[ 0 = u^2 + 2(-g)H \]Rearranging to solve for maximum height \(H\):\[ u^2 = 2gH \implies H = \frac{u^2}{2g} \]This equation establishes that maximum height is directly proportional to the square of the initial velocity (\(H \propto u^2\)).
Step 3: Derivation:
Let the initial velocities of the two stones be \(u_1\) and \(u_2\).The ratio of their velocities is provided as:\[ \frac{u_1}{u_2} = \frac{2}{5} \]Let the maximum heights reached by these stones be \(H_1\) and \(H_2\).Applying the formula \(H = \frac{u^2}{2g}\), the ratio of their heights is:\[ \frac{H_1}{H_2} = \frac{u_1^2 / (2g)}{u_2^2 / (2g)} = \frac{u_1^2}{u_2^2} = \left(\frac{u_1}{u_2}\right)^2 \]Substituting the given velocity ratio:\[ \frac{H_1}{H_2} = \left(\frac{2}{5}\right)^2 = \frac{4}{25} \]Step 4: Conclusion:
The ratio of the maximum heights achieved by the stones is 4:25.