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 $t_1t_2$ is

A particle is thrown with a velocity of $u \,m / s$. It passes $A$ and $B$ as shown in figure at time $t_1=1 \,s$ and $t_2=3 \,s$. The value of $u$ is ……. $m / s$ $\left(g=10 \,m / s ^2\right)$

Given below are two statements : one is labelled as Assertion $A$ and the other is labelled as Reason $R$.

Assertion $A$ : When a body is projected at an angle $45^{\circ}$, it's range is maximum.

Reason $R$ : For maximum range, the value of $\sin 2 \theta$ should be equal to one.

In the light of the above statements, choose the correct answer from the options given below :

A projectile is launched at an angle ' $\alpha$ ' with the horizontal with a velocity $20 \; ms ^{-1}$. After $10 s$, its inclination with horizontal is ' $\beta$ '. The value of $\tan \beta$ will be : $\left( g =10 \; ms ^{-2}\right)$

Figure shows four paths for a kicked football. Ignoring the effects of air on the flight, rank the paths according to initial horizontal velocity component, highest first