# Q.a) What is the trajectory followed by a projectile. b) What are the vertical and horizontal components of velocity? c) Derive an expression for time of flight.

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(a) Parabolic

(b) Vertical component is vsinθ and Horizontal component is vcosθ

(c)

1. Consider a body projected with velocity $\overset\rightarrow{u}$, at an angle θ of projection from point O in the coordinate system of the X-Y plane, as shown in figure.

2. The initial velocity $\overset\rightarrow{u}$ can be resolved into two rectangular components:

ux = ucos θ (Horizontal component)

uy = usin θ (Vertical component)

3. Thus, the horizontal component remains constant throughout the motion due to the absence of any force acting in that direction, while the vertical component changes according to, vy = uy + ayt with ay = – g and uy = u sinθ

4. The components of velocity of the projectile at time t are given by, vx = ux = u cos θ

vy = uy – gt = usinθ – gt ... (1)

5. At maximum height, vy = 0, t = tA = time of ascent = time taken to reach maximum height.

∴ 0 = usinθ - gtA ... (From 1)

∴ tA = usinθ / g

6. Total time of flight:-

T = 2tA = $\frac{2usinθ}{g}$