Motion in 3d - acceleration and its components

Question

Hi

 

I have found the vector for the acceleration of a particle in 3d, however I have been asked now to give its component that is tangential to the trajectory of the particle. Can I get some help with this please? Thanks

Answer 1

You need to find the unit tangent vector T(t), and evaluate the dot product of T(t) with your acceleration vector a(t).  That will give the tangential acceleration a_t(t) at time t.  I assume you started out with a position vector vector r(t), then you found a velocity vector

 

v(t) = dr(t)/dt,

 

and then an acceleration vector

 

a(t) = dv(t)/dt

 

If you already have dr(t)/dt, the unit tangent vector is just

 

T(t) = dr(t)/dt / |dr(t)/dt| = v(t) / |v(t)|

 

Then

 

a_t(t) = T(t) · a(t) = [v(t) / |v(t)|] · a(t)