Two particles are projected from a point at the same instant with velocities whose horizontal components and vertical components are $(u_{1}, v_{1})$ and $(u_{2}, v_{2})$ respectively such that there is another common point (other than origin) on their trajectories. The time interval between their passing through the other common point of their path (other than origin) is $\frac{1}{n} ∣ ∣ \frac{v_{1} u_{2} - v_{2} u_{1}}{u_{1} + u_{2}} ∣ ∣$ . Here n is an integer. Find n. $U s e g = 10 \frac{m}{s^{2}}$

Open in App

Solution

Let time interval between passing through another common point is $Δ t$ . Let first particle reaches there at time t and second particle at $t + Δ t$ . Let the common point has coordinate (x, y).

For $1^{s t}$ particle
$x = u_{1} t . . . . . . . (i)$
$y = v_{1} t - \frac{1}{2} g t^{2} . . . . . . . . (i i)$

For 2nd particle
$x = u_{2} (t + Δ t) . . . . . . . (i i i)$
$y = v_{2} (t + Δ t) - \frac{1}{2} g (t + t Δ +)^{2} . . . . . . . . . . (i v)$

From (i) & (iii)
$t = \frac{u_{2} Δ t}{u_{1} - u_{2}}$

From (ii) and (iv)
$v_{1} t - \frac{1}{2} g t^{2} = v_{2} (t + Δ t) - \frac{1}{2} g (t + Δ t)^{2}$
Substituting the value of $t$
$v_{1} \frac{u_{2} Δ t}{u_{1} - u_{2}} - \frac{1}{2} g (\frac{u_{2} Δ t}{u_{1} - u_{2}})^{2} = v_{2} (\frac{u_{2} Δ t}{u_{1} - u_{2}} + Δ t) - \frac{1}{2} g (\frac{u_{2} Δ t}{u_{1} - u_{2}} + Δ t)^{2}$

$v_{1} \frac{u_{2} Δ t}{u_{1} - u_{2}} - \frac{1}{2} g \frac{(u_{2} Δ t)^{2}}{(u_{1} - u_{2})^{2}} = v_{2} (\frac{u_{2} Δ t}{u_{1} - u_{2}} + Δ t) - \frac{1}{2} g \frac{(u_{1} Δ t)^{2}}{(u_{1} - u_{2})^{2}}$

$⟹ Δ t = \frac{2}{g} ∣ ∣ \frac{v_{1} u_{2} - v_{2} u_{1}}{u_{1} + u_{2}} ∣ ∣ = \frac{2}{10} ∣ ∣ \frac{v_{1} u_{2} - v_{2} u_{1}}{u_{1} + u_{2}} ∣ ∣ = \frac{1}{5} ∣ ∣ \frac{v_{1} u_{2} - v_{2} u_{1}}{u_{1} + u_{2}} ∣ ∣$ .
Thus, $n = 5$ .

Suggest Corrections

Similar questions

Q. Two particles are projected from a point at the same instant with velocities whose horizontal and vertical components are

u_{1}, v_{1}

and

u_{2}, v_{2}

respectively. The interval between their passing through the other common point of their path is

Q. Two particles are projected simultaneously in the same vertical plane from the same point, with different speeds

u_{1}

and

u_{2}

, making angles

θ_{1}

and

θ_{2}

respectively with the horizontal, such that

u_{1} cos θ_{1} = u_{2} cos θ_{2}

. The path followed by one, as seen by the other (as long as both are in flight) is:

A particle P is projected with velocity $u_{1}$ at an angle of $30^{\circ}$ with the horizontal. Another particle Q is thrown vertically upwards with velocity $u_{2}$ from a point vertically below the highest point of path of P. The necessary condition for the two particles to collide at the highest point is

Q. In the fig. shown below a body is projected with velocity

u_{1}

from the point A at the same time another body is projected vertically upwards with the velocity

u_{2}

from the point B. What should be the value of

u_{1}

u_{2}

for both the bodies to collide.

Join BYJU'S Learning Program