Question

A 5.00 kg particle starts from the origin at the time zero. its velocity as function of time is v=6t^2i+2tj where v in meters per second and t is in seconds.(a) find its position as a function of time.(b) describe its motion qualitatively.Find (c) its acceleration as a function of time, (d) the net force exerted on the particle as a function of time, (e) the net torque about the origin exerted on the particle as a function of time,(f) the angular momentum of the particle as a function of time,(g) the kinetic energy from the particle as a function of time, and (h) the power injected into the system of the particle as a function of time.

Answer 1

(a) The position as function of time is

x(t)=2t^3\hat\textbf i+t^2\hat\textbf j.

(b) Describe the motion qualitatively. Motion along i-ais (x-axis) is non-uniformly accelerated, motion along y-axis (j-axis) is uniformly accelerated.

(c) Acceleration as function of time:

a(t)=12t\hat \textbf i+2\hat\textbf j.

(d) The net force is

F(t)=ma(t)=60t\hat\textbf i+10\hat\textbf j.

(e) The net torque is

\tau(t)=Fx=120t^4\hat\textbf i+100t^2\hat\textbf j.

(f) The angular momentum is

L(t)=xp=xmv=60t^5\hat\textbf i+10t^3\hat\textbf j.

(g) The kinetic energy is

K(t)=\frac12mv^2(t)=90t^4\hat\textbf i+10t^2\hat\textbf j.

(h) The power is

P(t)=Fv=360t^3\hat\textbf i+20t\hat\textbf j.