A particle moves along a path described by y = Ax^3 and x= Bt where t is time What is the average velocity vector vt0 of this particle taken from time t = 0 to t = t0

To find the average velocity vector of the particle, we need to find the displacement vector and divide it by the time interval.

The displacement vector from t=0 to t=t0 is given by:

Δr = (x(t0) - x(0))i + (y(t0) - y(0))j

Substituting the given equations for x and y:

Δr = (Bt0 - 0)i + (A(Bt0)^3 - A(0)^3)j Δr = Bt0i + AB^3t0^3 j

The time interval is t0 - 0 = t0. So the average velocity vector is:

v(t0) = Δr / t0 v(t0) = (Bt0i + AB^3t0^3 j) / t0 v(t0) = B i + AB^3t0^2 j

Therefore, the average velocity vector of the particle from time t=0 to t=t0 is v(t0) = B i + AB^3t0^2 j