A rocket moves horizontally with a constant velocity u at a height 1. A gu lded missile is fired vertically with a speed v when the rocket passes above it. Assuming that the missile always aims at the rocket with the constant speed v, find the time after which the missile strikes the rocket.

Explanation

Let r=distance of separation between the rocket and missile after a time t. since, the missile always aims at the rocket with a constant speed v, it must move in a curve . At

time t if `vecr` makes an angle `theta` with horizontal , the missile approaches the rocket with `(v_("rel"))_(||)=(v-u cos theta)`. Hence, during a taken dt
`-dr=(v-cos theta)dt` .........(i)
since, at t=0,r=1 and at `t=tau,r=0`
`-int_(1)^(0)dr=int_(0)^(tau)(v- cos theta)dt`
or, `vtau-uint_(0)^(tau)cos theta=1` .....(ii)
Since, the missile hit the rocket, their x-displacements are same.
`Deltax_(M)=Deltax_(R)`
or, `int_(0)^(tau)v_(M)dt=int_(0)^(tau)v_(R)dt`
or, `int_(0)^(tau)v cos theta dt=int_(0)^(tau)udt`
or, `vintcos thetadt=utau` ......(iii)
and `utau=vint_(0)^(tau)cos thetadt`........(iv)
Eliminating `int_(0)^(tau)cos thetadt` from the Eqn. (ii) by substituting `int_(0)^(tau)cos theta dt=(u)/(v)tau` from the Eqn. (iv), we have
`tau=(vl)/(v^(2)-u^(2))`