Answer:
R = V₀² Sin 2θ₀/g
Step-by-step explanation:
Consider a projectile motion with following properties:
R = range of projectile
V₀ = Launch Velocity of Projectile
θ₀ = Launch Angle
V₀ₓ = V₀ Cos θ₀ = x - component of launch velocity
V₀y = V₀ Sin θ₀ = y - component of launch velocity
t = time to reach maximum height
T = 2t = total time of flight
First we use 1st equation of motion in vertical direction to find value of "t":
Vfy = V₀y + gt
where,
Vfy= final vertical velocity at highest point = 0 m/s
g = -g (for upward motion)
V₀y = V₀ Sin θ₀
Therefore,
0 = V₀ Sin θ₀ - gt
t = V₀ Sin θ₀/g
Now the total time of flight will be:
T = 2t = 2 V₀ Sin θ₀/g
Since, the horizontal velocity of the projectile remains uniform throughout the motion. Therefore:
x - x₀ = V₀ₓ T
where,
x - x₀ = R = Range of Projectile
V₀ₓ = V₀ Cos θ₀
Therefore,
R = (V₀ Cos θ₀)(2 V₀ Sin θ₀/g)
R = V₀² (2 Sin θ₀Cos θ₀)/g
since, 2 Sin θ Cos θ = Sin 2θ
Therefore,
R = V₀² Sin 2θ₀/g
