Name: Motion in a Plane With Constant Acceleration
Uploaded: 2019-07-19
Duration: 6 min 42 s
Description: In this video, we are going to look at the simple concept of motion in a plane with constant acceleration. You'll be able to master the concept in just a few...

Question

A rocket moves horizontally with a constant velocity $u$ at a height $h$ . A guided missile is fired vertically with a speed when the rocket passes above it.

Assuming that the missile always aims at the rocket with the constant speed $v$ , find the time after the missile strikes the rocket.

Accepted Answer

Consider at any instant the velocity vector of missile makes an angle $θ$ with the velocity vector of the rocket and after time $t$ missile strikes the rocket.

The approach velocity missile w.r.t. rocket can be written as

Velocity of $A$ w.r.t. $B$ at given instant $(v - u \cos θ)$

If $r$ is the distance of separation between rocket and missile after a time $t .$
$(v - u \cos θ) = - \frac{d r}{d t}$
$(v - u \cos θ) d t = - d r$

Integrating the above equation on both sides,

From $t = 0$ to $t = T$ and $r = h$ to $r = 0$ , We get

$\int_{0}^{T} v d t - \int_{0}^{T} u \cos θ d t = - \int_{h}^{0} d r$

$v T - u \int_{0}^{T} \cos θ d t = h . . . . . . . (1)$

The missile will strike the rocket when the displacement in the horizontal direction for missile and rocket is equal.

The velocity of missile in horizontal direction is $v \cos θ$ and for rocket the horizontal velocity is $u$ .

After time $d t$ both missile and rocket will cover the same displacement in the horizontal direction . So, we can write for time $t = 0$ to $t = T$

$\int_{0}^{T} v \cos θ d t = \int_{0}^{T} u d t v \int_{0}^{T} \cos θ d t = u T$
$\int_{0}^{T} \cos θ d t = \frac{u T}{v} . . . . . . . (2)$

From equation $(1)$ and $(2)$ ,We can write

$v T - u (\frac{u T}{v}) = h T = \frac{h v}{v^{2} - u^{2}}$

After time $T = \frac{h v}{v^{2} - u^{2}}$ missile will hit the rocket.

A rocket moves horizontally with a constant velocity $u$ at a height $h$ . A guided missile is fired vertically with a speed when the rocket passes above it.

Assuming that the missile always aims at the rocket with the constant speed $v$ , find the time after the missile strikes the rocket.

Important Questions on Kinematics II

Learn and Prepare for any exam you want !

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

Important Questions on Kinematics II

Learn and Prepare for any exam you want !

A rocket moves horizontally with a constant velocity $u$ at a height $h$ . A guided missile is fired vertically with a speed when the rocket passes above it.

Assuming that the missile always aims at the rocket with the constant speed $v$ , find the time after the missile strikes the rocket.