# Missile velocity kinematics

## Homework Statement

There is a target at $$r_0=x_0i+y_0j+z_0k$$ moving with a velocity of $$v=v_xi+v_yj+v_zk$$ wrt a missle launcher. With what velocity should the missile be launched so as to hit the target, and find the equation of trajectory of the missile.

## The Attempt at a Solution

I did this using relative velocity.

Let the velocity of the missile be $$v_m=vm_xi+vm_yj+vm_zk$$.

The relative velocity is $$v_{rel}=(v_x-vm_x)i+(v_y-vm_y)j+(v_z-vm_z)k$$.

The distance between them is $$r_0$$ and using equations of kinematics, $$r_0=v_{rel}t_0$$, where $$t_0$$ is the time to impact.

Now, here at least one of the two should be constant, either the speed of the missile or the time to impact. Assuming the time to impact to be constant, we get $$v_{rel}=\frac{r_0}{t_0}$$. Since we know the velocity of the target, we can find the velocity of the missile.

Integrating the velocity of the missile wrt t with limits 0 to $$t_0$$, we get the trajectory of the missile. Is this correct? Is there some other way to do this?

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nrqed
Homework Helper
Gold Member

## Homework Statement

There is a target at $$r_0=x_0i+y_0j+z_0k$$ moving with a velocity of $$v=v_xi+v_yj+v_zk$$ wrt a missle launcher. With what velocity should the missile be launched so as to hit the target, and find the equation of trajectory of the missile.
It's not clear form your question but I am assuming that the rocket launcher is at the origin and that there is no acceleration at all? (you treat the problem as if it was in space, far from any planet or star?)

I am assuming that the initial position of the target and its velocity are known? But all the components of the rocket velocity are unknown? You are not given the speed of thr rocket?

There is a range of possible answers then. Too many unknowns.

It's not clear form your question but I am assuming that the rocket launcher is at the origin and that there is no acceleration at all? (you treat the problem as if it was in space, far from any planet or star?)

I am assuming that the initial position of the target and its velocity are known? But all the components of the rocket velocity are unknown? You are not given the speed of thr rocket?

There is a range of possible answers then. Too many unknowns.
Yeah. Exactly. Except that rocket velocity is constant. Sorry I didnt point that out. Would this work though, as a general expression?

Like nrqed said, there are too many unknowns in the question. The way I read it, I can visualize a target moving with some specific velocity in cartesian space, and a missile(physics unknown) is fired at it from some location in space. I'm sorry but I can't do much with a question like this. Is this for some game simulation or something?

prasannapakkiam
well relative velocities won't exactly be the difference... Okay, the speed of the missile is considered to be low...

ANYWAY VECTORS CAN ONLY DENOTE MOVEMENT IN A DIRECTION AND MAGNETUDE. VECTORS DO NOT DEFINE ANY POINT IN SPACE. Although a general equation can be formed, the coordinates of the launcher and the target must be given. Otherwise, for all I know it could be anywhere... Also like the previous posters have said, an important variable is also the maximum velocity of the launcher can give the missile. If you are counting the sudden acceleration or any other acceleration of the missile, they must be given as this affects the angle or the direction in which you fire the missile...

well relative velocities won't exactly be the difference... Okay, the speed of the missile is considered to be low...

ANYWAY VECTORS CAN ONLY DENOTE MOVEMENT IN A DIRECTION AND MAGNETUDE. VECTORS DO NOT DEFINE ANY POINT IN SPACE. Although a general equation can be formed, the coordinates of the launcher and the target must be given. Otherwise, for all I know it could be anywhere... Also like the previous posters have said, an important variable is also the maximum velocity of the launcher can give the missile. If you are counting the sudden acceleration or any other acceleration of the missile, they must be given as this affects the angle or the direction in which you fire the missile...
As ngred pointed out and I confirmed, the launcher is at the origin. The velocity of the target is a constant. And I know about the vectors. There is no acceleration being taken into account. Everything is "as is". Its a pretty simple thing. Anyway, thanks a lot for your input.

nrqed
Homework Helper
Gold Member

## Homework Statement

There is a target at $$r_0=x_0i+y_0j+z_0k$$ moving with a velocity of $$v=v_xi+v_yj+v_zk$$ wrt a missle launcher. With what velocity should the missile be launched so as to hit the target, and find the equation of trajectory of the missile.

## The Attempt at a Solution

I did this using relative velocity.

Let the velocity of the missile be $$v_m=vm_xi+vm_yj+vm_zk$$.

The relative velocity is $$v_{rel}=(v_x-vm_x)i+(v_y-vm_y)j+(v_z-vm_z)k$$.

The distance between them is $$r_0$$ and using equations of kinematics, $$r_0=v_{rel}t_0$$, where $$t_0$$ is the time to impact.

Now, here at least one of the two should be constant, either the speed of the missile or the time to impact. Assuming the time to impact to be constant, we get $$v_{rel}=\frac{r_0}{t_0}$$. Since we know the velocity of the target, we can find the velocity of the missile.

Integrating the velocity of the missile wrt t with limits 0 to $$t_0$$, we get the trajectory of the missile. Is this correct? Is there some other way to do this?

I get almost the same equation.

Another derivation is :

position versus time of the missile:
$$\vec{r}_m = \vec{v}_m t$$

Position versus time of the target:

$$\vec{r}_t = \vec{r}_0 + \vec{v}_t t$$

There is an intercept when the two positions are equal at the same time, so we just set the two equal and isolate

$$\vec{r}_0 = (\vec{v}_m - \vec{v}_t) t$$
which has the opposite sign of your equation (your v relative is v_t minus v_m)

Hope this helps.

I get almost the same equation.

Another derivation is :

position versus time of the missile:
$$\vec{r}_m = \vec{v}_m t$$

Position versus time of the target:

$$\vec{r}_t = \vec{r}_0 + \vec{v}_t t$$

There is an intercept when the two positions are equal at the same time, so we just set the two equal and isolate

$$\vec{r}_0 = (\vec{v}_m - \vec{v}_t) t$$
which has the opposite sign of your equation (your v relative is v_t minus v_m)

Hope this helps.
Yes. Thank you. Any ideas on how to go about coding this?

nrqed
Homework Helper
Gold Member
Yes. Thank you. Any ideas on how to go about coding this?
Well, if I understand correctly you know the initial position r_0 and the velocity of the target. Just isolate the velocity of the missile:

$$\vec{v}_m = \frac{\vec{r}_0}{t} + \vec{v}_t$$

So you could just pick an arbitrary time "t" and plug it in to find the velocity you want.

You may also want tolook at the link provided by Dick.