Finding force from momentum vector

[Physics lover]

Homework Statement

The momentum of a particle is P vector=$\vec A$+($\vec B$×$t^2$),where $\vec A$ and $\vec B$ are constant perpendicular vectors.The force acting on the particle when its acceleration is at 45° with its velocity is

Relevant Equations

F=dP/dt

I calculated force vector by differentiating momentum vector.Since acceleration and velocity vectors are at45°,therefore force and momentum vector are at 45°.But i am not able to find the time at which it will take place.I tried F vector.P vector=FPcos45° but i am not getting from it.I also used that Avector.B vector=0.Help please.

 

[timetraveller123]

what did you get for force when you differentiated
$\vec P = \vec A + \vec B t^2$
and what did you get when you subbed it into this
$|\vec P||\vec F|cos 45 = \vec F . \vec P$

edit i did not try the first method
i was wondering
$\hat A , \hat B$
be the coordinate axis and since the force always aligned to one of them ...
this method seems lot faster than doing dot product

 

[Physics lover]

what did you get for force when you differentiated
$\vec P = \vec A + \vec B t^2$
and what did you get when you subbed it into this
$|\vec P||\vec F|cos 45 = \vec F . \vec P$

edit i did not try the first method
i was wondering
$\hat A , \hat B$
be the coordinate axis and since the force always aligned to one of them ...
this method seems lot faster than doing dot product

I got $\vec F$=2$\vec B$t
And by doing $\vec F$.$\vec P$ , a cubic power of t was coming and it was becoming complicated so i need help for the dot product.
And can you explain me which faster method are you talking about.

 

[timetraveller123]

I got $\vec F$=2$\vec B$t
And by doing $\vec F$.$\vec P$ , a cubic power of t was coming and it was becoming complicated so i need help for the dot product.
And can you explain me which faster method are you talking about.

well you know the force is always aligned with $\vec B$ and b and a are perpendicular so you can just call a the y-axis and b the x-axis so the force always points in the x-axis and is growing linearly in time
and for the momentum it starts out in the y-axis since at t=0 it is just a and as time increases it gets more and more component in the x-axis it becomes flatter and flatter and at some point it needs to be 45 degrees to the x axis(force ) what does that imply about the momentum at that point

for dot product
$\vec F = 2t \vec B\\ \vec P = \vec A + \vec B t^2\\ \vec F . \vec P = 2t^3 |B|^2 = \sqrt{A^2 + t^2 B^2}2t|B|\frac{1}{\sqrt{2}}$
many things cancel out