A boy playing on the roof of a 10 m high building throws a ball with a speed of $10 \mathrm{~m/s}$ at an angle of $30^{\circ}$ with the horizontal. How far from the throwing point will the ball be at the height of 10 m from the ground?
$$\left[\mathrm{g} = 10 \mathrm{~m/s}^{2}, \sin 30^{\circ} = \frac{1}{2}, \cos 30^{\circ} = \frac{\sqrt{3}}{2}\right]$$
(1) 5.20 m
(2) 4.33 m
(3) 2.60 m
(4) 8.66 m

A projectile can have the same range $R$ for two angles of projection. If $T _ { 1 }$ and $T _ { 2 }$ be the time of flights in the two cases, then the product of the two time of flights is directly proportional to
(1) $1 / R ^ { 2 }$
(2) $1 / R$
(3) R
(4) $R ^ { 2 }$

A projectile can have the same range R for two angles of projection. If $\mathrm{t}_1$ and $\mathrm{t}_2$ be the times of flights in the two cases, then the product of the two time of flights is proportional to
(1) $R^2$
(2) $1/R^2$
(3) $1/R$
(4) R

A shell is fired from a fixed artillery gun with an initial speed $u$ such that it hits the target on the ground at a distance $R$ from it. If $t_1$ and $t_2$ are the values of the time taken by it to hit the target in two possible ways, the product $t_1 t_2$ is:
(1) $R/2g$
(2) $R/g$
(3) $2R/g$
(4) $R/4g$

Two particles are projected from the same point with the same speed $u$ such that they have the same range $R$, but different maximum heights, $\mathrm { h } _ { 1 }$ and $\mathrm { h } _ { 2 }$. Which of the following is correct?
(1) $R ^ { 2 } = h _ { 1 } h _ { 2 }$
(2) $R ^ { 2 } = 4 h _ { 1 } h _ { 2 }$
(3) $R ^ { 2 } = 2 h _ { 1 } h _ { 2 }$
(4) $R ^ { 2 } = 16 h _ { 1 } h _ { 2 }$

The ranges and heights for two projectiles projected with the same initial velocity at angles $42 ^ { \circ }$ and $48 ^ { \circ }$ with the horizontal are $R _ { 1 } , \quad R _ { 2 }$ and $H _ { 1 } , \quad H _ { 2 }$ respectively. Choose the correct option:
(1) $R _ { 1 } = R _ { 2 }$ and $H _ { 1 } = H _ { 2 }$
(2) $R _ { 1 } = R _ { 2 }$ and $H _ { 1 } < H _ { 2 }$
(3) $R _ { 1 } > R _ { 2 }$ and $H _ { 1 } = H _ { 2 }$
(4) $R _ { 1 } < R _ { 2 }$ and $H _ { 1 } < H _ { 2 }$

An object is projected in the air with initial velocity $u$ at an angle $\theta$. The projectile motion is such that the horizontal range $R$, is maximum. Another object is projected in the air with a horizontal range half of the range of first object. The initial velocity remains same in both the case. The value of the angle of projection, at which the second object is projected, will be $\_\_\_\_$ degree.

Two projectiles are fired with same initial speed from same point on ground at angles of $(45^\circ - \alpha)$ and $(45^\circ + \alpha)$, respectively, with the horizontal direction. The ratio of their maximum heights attained is:
(1) $\frac{1 - \tan\alpha}{1 + \tan\alpha}$
(2) $\frac{1 - \sin 2\alpha}{1 + \sin 2\alpha}$
(3) $\frac{1 + \sin 2\alpha}{1 - \sin 2\alpha}$
(4) $\frac{1 + \sin\alpha}{1 - \sin\alpha}$