Acceleration always perpendicular to velocity

[randomafk]

Homework Statement

If the acceleration/force vector is always perpendicular to the velocity vector, what is the path?

Homework Equations

F=ma
a dot v=0

The Attempt at a Solution

We know that the dot product of a and v is zero such that

vx*vx'+vy*vy'=0 where vx'=dvx/dt

Also, I know this would be UCM, and given that I could say that the speed is constant. However, how would you know speed is constant not knowing the final path?

In addition, extending this further, how would know what path a particle took given acceleration and velocity vectors. That is to say, what if acceleration and velocity were not always perpendicular and also changed in magnitude with time?

 

[tiny-tim]

welcome to pf!

hi randomafk! welcome to pf!

… how would you know speed is constant not knowing the final path?

v.v' = 0

 

[randomafk]

hi randomafk! welcome to pf!


v.v' = 0

thanks for the welcome and help

if s=speed, then
s=sqrt(v.v)
How does this thing relate to v.v' = 0

Even if you knew that speed is constant, how could that demonstrate UCM?

 

[Quinzio]

thanks for the welcome and help

if s=speed, then
s=sqrt(v.v)
How does this thing relate to v.v' = 0

Even if you knew that speed is constant, how could that demonstrate UCM?

What happens to the velocity vector ?

 

[tiny-tim]

How does this thing relate to v.v' = 0

integrate

 

[randomafk]

The velocity vector should stay constant in magnitude but change direction

integrate

okay. so if i do an integral over dv

dv=a dt

\intv.dv = \int v.adt=C since v.a=0 for all t

I still can't figure how to link back to v.v=0

 

[tiny-tim]

(just got up :zzz: …)

d(v.v)/dt = 2v.dv/dt