Question

The range $R$ of a projectile is same when its maximum heights are ${h_1}$
and${h_2}$. What is the relation between$R$, ${h_1}$and${h_2}$?

Answer 1

Hint: The maximum height of the projectile along the horizontal plane is given is given by
${h_1} = \dfrac{{{u^2}{{\cos }^2}\theta }}{{2g}}$ and the maximum height of the projectile
along the vertical plane is given by ${h_2} = \dfrac{{{u^2}{{\sin }^2}\theta
}}{{2g}}$

Any object that is thrown or projected vertically upward when starts falling down gains a motion on its own inertia i.e. the tendency to remain constant. The motion is greatly influenced by the force of gravity. In simple wordS projectile has a single force which is the gravitational force acting upon it. Inertia can be defined as the property of an object that tends to remain in its state of rest or motion until and unless it is disturbed by a force from an external source.
The projectile motion has two types along the x-axis and along the y-axis.
Newton law of projectile motion also says that only force is required to produce acceleration but not the motion.
The projection of the projectile should be done at an angle of $\theta $ or $\left( {90^\circ -
\theta } \right)$ from the horizontal to the vertical.
Therefore, for these angles:

${h_1} = \dfrac{{{u^2}{{\sin }^2}\left( {90^\circ - \theta } \right)}}{{2g}} =
\dfrac{{{u^2}{{\cos }^2}\theta }}{{2g}}$ and ${h_2} = \dfrac{{{u^2}{{\sin }^2}\theta
}}{{2g}}$
Now, multiplying both the heights ${h_1}$ and ${h_2}$.
$\begin{array}{c}{h_1}{h_2} = \dfrac{{{u^2}{{\cos }^2}\theta }}{{2g}} \times
\dfrac{{{u^2}{{\sin }^2}\theta }}{{2g}}\\ = \dfrac{{{u^4}{{\sin }^2}\theta {{\cos }^2}\theta
}}{{4{g^2}}}\\ = \dfrac{{{u^4}4{{\sin }^2}\theta {{\cos }^2}\theta }}{{4 \times 4{g^2}}}\\ =
\dfrac{1}{{16}}{\left( {\dfrac{{{u^2}\sin 2\theta }}{g}} \right)^2}\end{array}$
Simplify the above equation.
$16{h_1}{h_2} = {\left( {\dfrac{{{u^2}\sin 2\theta }}{g}} \right)^2}$ ……
(1)
Maximum range of the projectile is
\[R = \left( {\dfrac{{{u^2}\sin 2\theta }}{g}} \right)\] …… (2)
From equation (1) and equation (2),
$\begin{array}{c}16{h_1}{h_2} = {R^2}\\R = 4\sqrt {{h_1}{h_2}} \end{array}$
Few examples of projectile motion are-
3. The motion of a bullet fired from a gun.
4. A ball thrown from a particular height.
Note: The motion of a projectile is directly affected by the force of gravity, resistance made by the air, releasing speed of the projectile and the angle.