Question

Derive formulae for time of flight (T), maximum height (H) and horizontal range (R) of projectile motion.

Answer 1

Time of Flight

It is the total time taken by the projectile when it is projected from a point and reaches the same horizontal plane or the time for which the projectile remains in the air above the horizontal plane.
It is denoted by T.

As the motion from the point O to A and then from the point A to B are symmetrical, the time of ascent (for journey from point O to A) and the time of descent (for the return journey from A to B) will be each equal to T/2.

At the highest point A, the vertical component of velocity of the object becomes zero. Taking vertically upward motion of the object from O to A, we have
u_y = u sinθ, a_y = -g, t = \(\frac{T}{2}\) and v_y = 0
since, v_y = u_y + at
∴ 0 = u sinθ – g \(\frac{T}{2}\)
⇒ T = \(\frac{2 u \sin \theta}{g}\) …………….. (4)

Maximum height of a projectile
It is the maximum vertical height attained by the object above the point of projection during its flight. It is denoted by H.

Taking the vertical upward motion of the object 122 (B) from O to A, we have :

Horizontal-range
It is the horizontal distance covered by the object between its point of projection and the point of hitting the ground. It is denoted by R.

Obviously, the horizontal range R is the horizontal distance covered by the projectile with the’ uniform velocity u cosθ in a time equal to the time of flight.