How Does Acceleration Affect Constant Velocity in Particle Motion?

[Destrio]

A particle is moving in the xy plane with velocity v(t) = Vx(t)i + Vy(t)j
and acceleration a(t) = ax(t)i + ay(t)j
By taking the appropriate derivative, show that the magnitude of v can be constant only if axvx + ayvy = 0

I did it in a similar way to the other problem I had as it seems like a similar problem,
My thinking is:

h = (Vx(t)i)^2 + (Vy(t)j)^2

s(t) = sqrt(h)

dh/dt = d(vx^2 + vy^2) / dt
= d(vx)^2/dt + d(vy)^2/dt
= (d(vx^2)/dvx)(dvx/dt) + (d(vy^2)/dvy)(dy/dt)
= 2vx(dx/dt) + 2vy(dy/dt)

ds/dt = (ds/dh)(dh/dt)
= ((1/2)h^(-1/2))[(2vx(dvx/dt))+(2vy(dvy/dt))]

((1/2)h^(-1/2))[(2vx(dvx/dt))+(2vy(dvy/dt))] = 0

((1/2)h^(-1/2))[(2vx(ax))+(2vy(ax))] = 0

2((1/2)h^(-1/2))[vxax+vyax] = 0

axvx + ayvy = 0


Thanks

 

[learningphysics]

Looks good! Very nice!