Example 3 - Using Derivatives

Problem: The position of a particle is given by

Find:

a) the initial position of the particle

b) the position at t = 3 seconds

c) the velocity of the particle after 2 seconds

d) the acceleration of the particle at t = 2 seconds

e) could the constant acceleration formulas be used?

Solution:

a) Substituting t = 0 into the position vector, I get

b) Substituting t = 3 into the position vector, I get

(Note: The "3" in parentheses is functional notation; it is not multiplication.)

c) To find the velocity at any time, I take the derivative of the position vector:

Now I substitute t = 2 seconds to get

(Both notations are used.)

d) First, I find the general acceleration and then substitute 2 seconds. To take the derivative, the general velocity expression must be used, not the velocity at t = 2 seconds!

e) The constant acceleration formulas cannot be used, because the acceleration is a function of time!