Question

Ratio between the maximum range and square of time of flight in projectile motion is

Answer 1

the maximum range of projectile motion is given by the formula ,R= U²sin2theta/g.
and formula for T=2usin theta/g

then ,T²=4U²sin²theta/g²

so, for max. range sintheta =1
R .max.=U²/g
therefore,, required ratio is u²/g÷4u²sin²theta/g² .......

Answer 2

For maximum range θ = 45°
R_max = u² Sin(2θ) / g
= u² Sin90 / g
= u² / g

Then,
Time of flight
T = 2uSinθ / g
= 2u Sin45 / g
= 2u / (g√2)

Square of time of flight
T² = 4u² / (2g²)
= 2 u²/g²

Ratio....,
R_max / T² = [u² / g] / [2u² / g²]
= g / 2
= 10 / 2
= 5

Ratio is 5 : 1
[Note: I took g = 10 m/s^2. If g = 9.8 m/s^2 then ratio will be 4.9 : 1]