Question

What is projectile motion. Derive the expretion for height , range, and time of flight

Answer 1

Projectile motion is a 2-dimensional motion that has both x (horizontal ) and y (vertical ) direction in its motion.

Maximum Height: When a body is thrown upwards, the velocity of the body decreases and finally becomes zero (v = 0 m/s) at a certain height. This height is called the maximum height (denoted by H or Hmax)

From, v² = u² - 2gh

=> $0 = u^2 - 2gh_{max}$

=> $\fbox{h_{max} \;or \;H = \frac{u^2}{2g}}$

Time of flight:

Time of ascent: It is the time taken by a body, projected vertically upwards to reach the maximum height, So the final velocity (v) = 0 m/s

From the equation: v = u + at

=> 0 = u - gt$_a$

(where, a = -g, Since the acceleration is towards the gravity)

=> $\large{t_a = \frac{u}{g}}$

Time of descent: It is the time taken by a body, projected vertically upwards to return and come back to the point of projection from the maximum height, So the initial velocity (u) = 0 m/s

From the equation:

s = ut + $\frac{1}{2}gt^2$

=> $h_{max} = 0 + \frac{1}{2}gt_d\,^2$

=> $t_d = \sqrt{\frac{h<em>_</em><em>{</em><em>m</em><em>a</em><em>x</em><em>}</em> × 2}{g}}$

$As, h<em>_</em><em>{</em><em>m</em><em>a</em><em>x</em><em>}</em> = \frac{u^2}{2g}$

=> $t_{d} = \sqrt{\frac{u^2}{2g}×\frac{2}{g}}$

=> $t_d = \sqrt{\frac{u^2}{g^2}}$

=> $t_d = \frac{u}{g}$

Time of flight: It is the total time taken by a body projected vertically upwards to reach the position of maximum height and then return to the point of projection. (denoted by T)

Time of flight = $t_a + t_d$

=> T = $\frac{u}{g} + \frac{u}{g}$

=> $\large\fbox {T = \frac{2u}{g}}$

Range: The horixontal distance covered by a projectile motion is called Range, 'R' which it covers in a time 't' equal to time of flight,$t_{f}$, Thus,

R = u cosθ × $t_{f}$

The time of flight $t_{f}$ is the time taken by the projectile to reach the horizontal plane after being projected. It is given by,

$t_{f} = \frac{2u sinθ}{g}$

$So, R = u cosθ × \frac{2u sinθ}{g}$

$=> R = \frac{u^2(2sinθcosθ)}{g}$

$=> \fbox{R = \frac{u^2 sin2θ}{g}}$