Question

Galileo in his book. Two new sciences stated that “for elevations which exceed or fall short of ${45^ \circ }$ by equal amounts, the ranges are equal”. Prove this statement.

Answer 1

1. Elevation: It is the action of raising a body to a higher position against the activity of gravity.
2. Fall: It is the action of accelerating a body downward due to the activity of the gravity.
3. Angle of repose: It is the minimum angle required to slide down an object placed on an inclined plane, the angle between the inclined plane and the horizontal which causes the sliding of the body is called the angle of inclination.

Complete step by step solution:
Parabolic motion: It is the motion due to gravity, when an object is thrown in an upward direction the body falls on the ground under the action of gravity. In the horizontal direction, the body travels at a constant speed during the flight.
For elevation: The angle is$({45^ \circ } + \theta )$, the range is calculated as, ${{\text{R}}_1} = \dfrac{{{{\text{v}}_0}^2\sin [2({{45}^ \circ } + \theta )]}}{{\text{g}}}$
Here, v0= velocity of the projectile motion, g= acceleration due to gravity
${{\text{R}}_1} = \dfrac{{{{\text{v}}_0}^2\sin ({{90}^ \circ } + 2\theta )}}{{\text{g}}}$ $= \dfrac{{{{\text{v}}_0}^2\cos 2\theta }}{{\text{g}}}$, here, ${\text{sin(90 + 2}}\theta {\text{) = cos2}}\theta$
For fall: The angle is $({45^ \circ } - \theta )$, the range is calculated as, ${{\text{R}}_2} = \dfrac{{{{\text{v}}_0}^2\sin [2({{45}^ \circ } - \theta )]}}{{\text{g}}}$
${{\text{R}}_2} = \dfrac{{{{\text{v}}_0}^2\sin ({{90}^ \circ } - 2\theta )}}{{\text{g}}}$ $= \dfrac{{{{\text{v}}_0}^2\cos 2\theta }}{{\text{g}}}$, here, ${\text{sin(90 - 2}}\theta {\text{) = cos2}}\theta$
From the above two equations, we get that, ${{\text{R}}_1} = {{\text{R}}_2}$
This is the required proof for the statement.

Note:
1. Range of the projectile motion depends on the acceleration due to gravity, velocity, acceleration, and time.
2. Angle of release, speed of release, the height of release are the primary factors affecting the projectile motion.
3. The firing of water, water jet from a hose, a long-distance shot of the golf ball are some examples of projectile motion.
4. When the object is to shoot directly in a downward direction, the minimum velocity will be equal to the initial velocity.
5. When the object is shooting in an upward direction the minimum velocity of the projectile motion will be at the peak point.