Velocity and Acceleration Components

Question:

How do you find the velocity components Vx(t) and Vy(t)? How do you find the acceleration components Ax(t) and Ay(t)? The vector position of a particle varies in time according to the expression r= (3.00i-6.00t2j)m.

Answer:

In general the approach to get velocity and acceleration as a function of time if you know position as a function of time is to differentiate position with respect to time to get velocity and differentiate velocity with respect to time to get acceleration. In your function of position vs time I assume i to be the unit vector along the x axis, j to be the unit vector along the y axis and m to be the unit of length, meters.

The derivative of a vector with respect to time is the vector whose components are the derivatives of the components of the original vector. So you may find the x component of velocity by taking the first derivative of 3.00 with respect to time. Since 3.00 is independent of time, the first derivative is zero. Vx(t)=0. Vy(t) is the first derivative of -6t2 or -12t, so Vy(t)=-12t. The acceleration components are the first derivatives of the velocity components. Here since Vx(t)=0 also Ax(t)=0. Ay(t) is the first derivative of -12t, or -12, Ay(t)=-12.

Reply if you need further explanation.

JDJ