Question

Two particles A and B are projected with same speeds so that ratio of their maximum heights reached is $3:1$. If the speed of A is doubled without altering other parameters, the ratio of horizontal ranges attained by A and B is:
(A) $1:1$
(B) $2:1$
(C) $4:1$
(D) $3:2$

Answer 1

In the question the ratio of maximum height and change in speed is given, so we can use the formula for maximum height and use the common terms in the equation of range of a projectile motion to relate it to the range and get the desired ratio.

Formula used:
$H = \dfrac{{{u^2}{{\sin }^2}\theta }}{{2g}}$
$R = \dfrac{{{u^2}\sin 2\theta }}{g}$

Complete step by step answer:
It is given that A and B will have projectile motion and the ratio of maximum height is $3:1$
We know the formula for maximum height, H in a projectile motion
$H = \dfrac{{{u^2}{{\sin }^2}\theta }}{{2g}}$where u is the initial velocity $\theta$is angle of projection and g is acceleration due to gravity
So, according to question $\dfrac{{{H_1}}}{{{H_2}}} = \dfrac{{{u_1}^2{{\sin }^2}{\theta _1}}}{{2g}} \div \dfrac{{{u_2}^2{{\sin }^2}{\theta _2}}}{{2g}} = \dfrac{{{u_1}^2{{\sin }^2}{\theta _1}}}{{{u_2}^2{{\sin }^2}{\theta _2}}}$
On substituting the given values,
$\dfrac{{{H_1}}}{{{H_2}}} = \dfrac{{{u^2}{{\sin }^2}{\theta _1}}}{{{u^2}{{\sin }^2}{\theta _2}}} \Rightarrow \dfrac{3}{1} = \dfrac{{{{\sin }^2}{\theta _1}}}{{{{\sin }^2}{\theta _2}}} \Rightarrow \dfrac{{\sin {\theta _1}}}{{\sin {\theta _2}}} = \sqrt {\dfrac{3}{1}}$
To get the value of the angle multiply and divide by 2,
Hence ${\theta _2} = {30^0},{\theta _1} = {60^0}$
Now let us write the formula of range $R = \dfrac{{{u^2}\sin 2\theta }}{g}$, so the ratio of range will be
$\dfrac{{{R_1}}}{{{R_2}}} = \dfrac{{{u_1}^2\sin 2{\theta _1}}}{{{u_2}^2\sin 2{\theta _2}}} = \dfrac{{4{u^2}\sin {{120}^0}}}{{{u^2}\sin {{60}^0}}} \Rightarrow \dfrac{{{R_1}}}{{{R_2}}} = \dfrac{{4\dfrac{{\sqrt 3 }}{2}}}{{\dfrac{{\sqrt 3 }}{2}}} = 4:1$

Hence, the correct option is C.

Additional information:
Projectile motion is a combination of horizontal and vertical motion which are independent of each other and are in mutually perpendicular direction.

Note:
While doing projectile motion questions we need to follow some assumptions like –
1. Assume that earth is flat. We will ignore the effect of air resistance on the body. Generally, we consider projectile motion of short range so that we can assume that gravitational force is constant.
2. We need to keep in mind that during a projectile motion, when the particle is moving upward the vertical component of velocity is positive and when the particle is moving downwards then the vertical component is negative. The formula of range and height used is invalid if the particles do not land at the same horizontal level as of the projection.