Derive R = v0 2 sin2θ0 / g for the range of a projectile on level ground by finding the time t at which y becomes zero and substituting this value of t into the expression for x − x0, noting that R = x − x0.

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

b

step-by-step explanation:

the word superfluous means unnecessary