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 \(15^{\circ},\) \(30^{\circ},\)\(45^{\circ},\) and \(60^{\circ}\) with the horizontal respectively. The body having the shortest range is? 

A stone projected with a velocity \(u\) at an angle \(\theta\) with the horizontal reaches maximum height \(H_1.\) When it is projected with velocity \(u\) at an angle \(\left(\frac{\pi}{2}-\theta\right)\) with the horizontal, it reaches maximum height \(H_2.\) The relation between the horizontal range of the projectile \(R\) and \(H_1\) and \(H_2\) is: 

Which of the following sets of factors will affect the horizontal distance covered by an athlete in a long–jump event?
1. speed before he jumps and his weight
2. the direction in which he leaps and the initial speed
3. the force with which he pushes the ground and his speed
4. none of the above

In a projectile motion, velocity at maximum height is 

(1) $\frac{u\cos \theta }{2}$

(2) $u\cos \theta$

(3) $\frac{u\sin \theta }{2}$

(4) None of these

The equation of motion of a projectile is given by x = 36 t metre and 2y = 96 t – 9.8 t² metre. The angle of projection is:

1. ${\sin }^{-1}\left(\frac{{\displaystyle 4}}{{\displaystyle 5}}\right)$

2. ${\sin }^{-1}\left(\frac{{\displaystyle 3}}{{\displaystyle 5}}\right)$

3. ${\sin }^{-1}\left(\frac{{\displaystyle 4}}{{\displaystyle 3}}\right)$

4. ${\sin }^{-1}\left(\frac{{\displaystyle 3}}{{\displaystyle 4}}\right)$

For a given velocity, a projectile has the same range of R for two angles of projection. If t₁ and t₂ are the times of flight in the two cases then:

(1) $t_1{t}_2\propto R^2$

(2) $t_1{t}_2\propto R$

(3) $t_1{t}_2\propto \frac{1}{R}$

(4) $t_1{t}_2\propto \frac{1}{R^2}$

A body of mass m is thrown upwards at an angle θ with the horizontal with velocity v. While rising up the velocity of the mass after t seconds will be 

(1) $\sqrt{{\left(v\cos \theta \right)}^2+{\left(v\sin \theta \right)}^2}$

(2) $\sqrt{{(v\cos \theta -v\sin \theta )}^2-gt}$

(3) $\sqrt{v^2+g^2{t}^2-\left(2v\sin \theta \right)gt}$

(4) $\sqrt{v^2+g^2{t}^2-\left(2v\cos \theta \right)gt}$ 

A cricketer can throw a ball to a maximum horizontal distance of \(100~\text{m}\). With the same effort, he throws the ball vertically upwards. The maximum height attained by the ball is: 
1. \(100~\text{m}\)
2. \(80~\text{m}\)
3. \(60~\text{m}\)
4. \(50~\text{m}\)