Question

4. A man wishes to throw two darts one by one at the target at B with same speed so that they arrive at the same time. Mark the correct statements about the two projections. Trajectory B Trajectory A А. B (A) Projectile that travels along trajectory A was projected earlier (B) Projectile that travels along trajectory B was projected earlier. (C) Second dart must be projected at angle e, such that 0 + 0,8 = 90° (D) Second dart must be projected at angle , >, STA​

Answer 1

The initial speed of the two darts are same.

• $\smallu_A=u_B=u$

The darts are projected from same point A but not at same time; they are thrown one by one. But they travel same horizontal distance and reach B at the same time.

Since the horizontal range of both darts are same,

$\small\longrightarrow\dfrac{u^2\sin(2\theta_A)}{g}=\dfrac{u^2\sin(2\theta_B)}{g}$

$\small\longrightarrow\sin(2\theta_A)=\sin(2\theta_B)$

Since $\small\theta_A\neq\theta_B,$

$\small\Longrightarrow2\theta_A+2\theta_B=180^o$

$\small\text{ \longrightarrow\underline{\underline{\theta_A+\theta_B=90^o}} }$

In the figure we see,

$\small\longrightarrow\theta_B>\theta_A$

Since both are acute angles,

$\small\longrightarrow\sin(\theta_B)>\sin(\theta_A)$

Multiplying both sides by $\small\dfrac{2u}{g}$ which is positive,

$\small\longrightarrow\dfrac{2u\sin(\theta_B)}{g}>\dfrac{2u\sin(\theta_A)}{g}$

That is,

$\small\longrightarrow T_B>T_A$

It means projectile that travels through trajectory B should travel more time than the projectile that travels through trajectory A, so it should be projected earlier in order to arrive at B at the same time.

Hence options (B) and (C) are correct.