# Math Help - Integrating to get velocity. (notation)

1. ## Integrating to get velocity. (notation)

I'm stuck on how to get the proper notation for this problem.

In physics, we know that Force= Mass x acceleration, or F=ma

In this case, the force is equal to $(A)/c * cos(theta)$

So i'm looking at $((A)/c * cos(theta))/m = a$

BUT, I need to velocity as a function of time, v(t).

Basically, what would a formula for $v(t)=$ look like?

2. Originally Posted by elninio
I'm stuck on how to get the proper notation for this problem.

In physics, we know that Force= Mass x acceleration, or F=ma

In this case, the force is equal to $(A)/c * cos(theta)$

So i'm looking at $((A)/c * cos(theta))/m = a$

BUT, I need to velocity as a function of time, v(t).

Basically, what would a formula for $v(t)=$ look like?
You could start by saying $a = \frac {dv}{dt}$ (rate of change of velocity with time) and so:

$(A)/c * cos(theta) = \frac {dv}{dt}$

If the expression on the left has no time dependency, i.e. is constant w.r.t. time, you can just write:

$t (A)/c * cos(theta) = v + v_0$

where $v_0$ is some arbitrary constant velocity that will be determined by applying some boundary condition or initial value or whatever.

3. So, If I wanted to find the Kinetic Energy, which equals 1/2mv^2, would that give me:

$KE=.5m(A(dt)/cm)^2$?

I hope you can follow my notation. (also, lets ignore cos theta)

4. Oh okay then ...

For a start I HATE that $.5 m$ - I much prefer $m/2$ at this stage, because after you've integrated with respect to $t$ your complicated expression with all that $ A$ in it (haven't a clue what it means BTW) you may find the 2 cancels out with something else.

Okay, so yes you get your velocity by integrating your LHS, then square it, and times it by $m/2$.

There's obviously something in the problem you're doing that you're not telling us ... we may be able to enlighten you better if you post the whole thing you're trying to solve.

5. Ok, here's the entire question. Its a high level astrophysics problem but I just needed a hand with the calculus part:

Consider a spacecraft of mass m whose engine is a perfectly absorbing laser sail that is initially at rest in space. No gravity is being exerted on it. We aim a laser at the sail and cause it to accelerate into deep space.

Derive a formula for its kinetic energy, as a function of its mass and the total amount of energy Ei that it absorbs from the laser beam during some time interval t. Assume that the spacecrafts velocity remains NON-relativistic.

I.E. You're firing radiation and using the pressure of light to accelerate it.

KE=1/2mv^2

6. You'd need to explain what S, A, c and theta are (although I guess c is the velocity of light). And what's the $$ notation? I'm a bit of a pure mathematician, physics notation I've never got on with, it tends to confuse me.

7. You can ignore those. <S> and A relate to the pointing vector. But in this situation, I am sure that they could be anything and the answer would turn out the same. It is unimportant what they mean and I just want to make sure I have my notation correct.
Consider the entire F(rad) formula a constant.

8. In that case, (shrug) I suppose it is ...