Two stones are thrown vertically upwards simultaneously with their initial velocities u1and u2 respectively. Prove that the heights reached by them would be in the ratio of u12 : u22 (Assume upward acceleration is –g and downward acceleration to be +g ).

The height reached by any stone can be calculated by using the third equation of motion.

$v^2=u^2+2as$

We can use the fact, that at the highest point the velocity will be zero. The acceleration due to gravity will retard the particles.

Therefore, the height raised for any particle will be given as:

$\\0^2=u^2+2as\\\Rightarrow h=\frac{u^2}{2g}$

Therefore

$\frac{h_1}{h_2}=\frac{u_1^2}{u_2^2}\\$