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If two bodies are projected with the same speed but at angles $\mathrm{}$\((45^\circ +\theta  )\) and $\mathrm{}$\((45^\circ -\theta  )\) repectively. Then their horizontal ranges are in the ratio of:
(If \(𝜃   ≤   45^\circ \))$\left(\right)$
1. \(2 : 1\)
2. \(1 : 2\)
3. \(1 : 1\)
4. \(2 : 3\)

For a given velocity, a projectile has same range for two angles of projection. If T₁ & T₂ are the times of flight in the two cases then
1. $T_1{T}_2 \propto \frac{1}{R}$
2. $T_1{T}_2\propto \frac{1}{R^2}$
3. $T_1{T}_2\propto R$
4. $T_1{T}_2\propto R^2$

If horizontal range of a projectile is 4$\sqrt{3}$ times its maximum height. Its range of projection will be
1. $45°$
2. $60°$
3. $90°$
4. $30°$

A stone is pfojected with a velocity v at an angle $\theta$ with the horizontal reaches maximum height H₁. When it is projected with u at an angle $\left(\frac{\mathrm{\pi }}{2}-\theta \right)$ with the horizontal, it reaches maximum height H₂. The ratio between hrizontal range  R of the projection, H₁ and H₂ is ?
1. $R = 2\sqrt{H_1{H}_2}$
2. $R = \sqrt{H_1{H}_2}$
3. $R = 4\sqrt{H_1{H}_2}$
4. $R = 8\sqrt{H_1{H}_2}$

The velocity of a projectile at the initial point A is $\left(4\hat{i}+5\hat{j}\right)$ m/s. Its velocity in m/s at point B is

1. $\left(4\hat{i}+5\hat{j}\right)$
2. $\left(-4\hat{i}-5\hat{j}\right)$
3. $\left(-4\hat{i}+5\hat{j}\right)$
4. $\left(4\hat{i}-5\hat{j}\right)$

A body is projected with initial kinetic energy of 100 J and at an angle of $60°$ with the horizontal. The K.E. of the body at highest point will be
1. 25 J
2. 50 J
3. 75 J
4. 0 J

The equation of trajectory of a projectile given angular projection is y = ax - bx², then horizontal range of projectile is 
1. $\frac{a}{b}$
2. $\frac{a}{2b}$
3. $\frac{2a}{b}$
4. $\frac{a}{4b}$

A large number of bullets are fired in all directions with same velocity u. What is the maximum area on the ground on which these bullets can spread ?
1. $\frac{{\mathrm{\pi u}}^2}{g}$
2. $\frac{{\mathrm{\pi u}}^4}{g^2}$
3. $\frac{{\pi }^2{u}^4}{g^2}$
4. $\frac{{\pi }^2{u}^2}{g^2}$

A particle is thown with a speed u at an angle $\theta$ with the horizontal. When the particle makes an angle $\varphi$ with the horizontal, its speed changes to v, Which is equal to 
1. $u \cos \theta$
2. $u \cos \theta \cos \varphi$
3. $u \cos \theta sec \varphi$
4. $u sec \theta \cos \varphi$

A smooth square platform ABCD is moving towards right with uniform speed u. At what angle $\theta$ must a particle be projected from A with speed v, so that it strikes the point D ?

1. ${\sin }^{-1}\left(\frac{u}{v}\right)$
2. ${\cos }^{-1}\left(\frac{v}{u}\right)$
3. ${\sin }^{-1}\left(\frac{v}{u}\right)$
4. ${\cos }^{-1}\left(\frac{u}{v}\right)$

Time taken by the projectile to reach from P to Q is t, then PQ ?
1.  2ut
2. $\sqrt{3 }ut$
3. $\frac{\sqrt{3}}{2}ut$
4. $\frac{ut}{\sqrt{3}}$

A aeroplane moving with a speed of 250m/s is at a height of 6000 m just overhead of an ant, aircraft gun. Muzzle speed of shell is 500m/s, the firing angle $\theta$ should be ?

1. ${45}^0$
2. ${60}^0$
3. ${30}^0$
4. None