Galileo writes that for angles of projection of a projectile at angles $(45+\theta )$ and $(45-\theta )$, the horizontal ranges described by the projectile are in the ratio of (if $\theta \le 45$) 

1. 2 : 1

2. 1 : 2

3. 1 : 1

4. 2 : 3

A projectile thrown with a speed v at an angle θ has a range R on the surface of earth. For same v and θ, its range on the surface of moon will be (acceleration due to gravity on moon=$\frac{g}{6}$):

(1) R/6

(2) 6 R

(3) R/36

(4) 36 R

A ball is projected with kinetic energy E at an angle of 45° to the horizontal. At the highest point during its flight, its kinetic energy will be 

(1) Zero

(2) $\frac{E}{2}$

(3) $\frac{E}{\sqrt{2}}$

(4) E

At the top of the trajectory of a projectile, the magnitude of the acceleration is

(1) Maximum

(2) Minimum

(3) Zero

(4) g

A body is projected at such an angle that the horizontal range is three times the greatest height. The angle of projection is 

(1) ${25}^o{8}^{\text{'}}$

(2) ${33}^o{7}^{\text{'}}$

(3) ${42}^o{8}^{\text{'}}$

(4) ${53}^o{8}^{\text{'}}$

Two bodies are projected with the same velocity. If one is projected at an angle of 30° and the other at an angle of 60° to the horizontal, the ratio of the maximum heights reached is 

(1) 3 : 1

(2) 1 : 3

(3) 1 : 2

(4) 2 : 1

If the range of a gun that fires a shell with muzzle speed v is R, then the angle of elevation of the gun is 

(1) ${\cos }^{-1}\left(\frac{{\displaystyle v^2}}{{\displaystyle Rg}}\right)$

(2) ${\cos }^{-1}\left(\frac{{\displaystyle gR}}{{\displaystyle v^2}}\right)$

(3) $\frac{1}{2}\left(\frac{{\displaystyle v^2}}{{\displaystyle Rg}}\right)$

(4) $\frac{1}{2}{\sin }^{-1}\left(\frac{{\displaystyle gR}}{{\displaystyle v^2}}\right)$

If a body A of mass M is thrown with a velocity v at an angle of 30° to the horizontal and another body B of the same mass is thrown with the same speed at an angle of 60° to the horizontal. The ratio of horizontal range of A to B will be 

(1) 1 : 3

(2) 1 : 1

(3) $1:\sqrt{3}$

(4) $\sqrt{3}:1$

Four bodies P, Q, R and S are projected with equal velocities having angles of projection 15^o, 30^o, 45^o and 60^o with the horizontal respectively. The body having the shortest range is? 

A stone projected with a velocity u at an angle θ with the horizontal reaches maximum height H₁. When it is projected with velocity u at an angle $\left(\frac{\pi }{2}-\theta \right)$with the horizontal, it reaches maximum height H₂. The relation between the horizontal range of the projectile R and H₁ & H₂ is: