Definition: If r(t) is a vector-valued function of t representing position as a function of time, t, then the average velocity of the particle over the time interval [t1,t2] is given by
[ r(t2) - r(t1) ]/[ t2 - t1 ] Note: the average velocity is a vector quantity!Definition: If r(t) is a vector-valued function of t representing position as a function of time, t, then the instantaneous velocity of the particle at t1 is given by
lim t2 -> t1 [ r(t2) - r(t1) ]/[ t2 - t1 ] Definition: The speed of the particle at time t is given by
speed = |r'(t)| Note: The speed is a scalar quantity, and is always non-negative!
Definition: The acceleration is the rate of change of velocity and is given by
acceleration = r''(t) Note: The concept of acceleration is an important one in physics. If the mass of an object remains constant, then
mass x acceleration = F external
Uniform circular motion in the plane. In this case we have x(t) = R cos(wt), y(t)=R sin(wt), where w is the angular velocity. Then
r (t) = < x(t),y(t) > = < R cos(wt), R sin(wt) > The velocity is given by d/dt r(t),
r'(t) = < -w R sin(wt), w R cos(wt) > Note: this is orthogonal to r(t)!
The acceleration is given by r''(t)
r''(t) = < -w2 R cos(wt), - w2 R sin(wt) > -w2 r(t) Note: This is in the opposite direction to r(t)!