2D Motion

We take the derivative of a vector-valued function coordinate by coordinate:

The first order of business is to write the above equations in vector form. Because they are all (at most quadratic) polynomials in t , we can write them together as:

We are now in a position to compute the velocity and acceleration functions. Using the rules established in this section we find that,

Notice that the acceleration function a(t) is constant; therefore the magnitude (and direction!) of the acceleration vector will be the same at all times:

All that's left now is to compute the magnitudes of the position and velocity vectors at times t = 0, 2, - 2 :

• At t = 0 , |x(0)| = |(5, -2, 1)| = , and |v(0)| = |(0, 3, 2)| =
• At t = 2 , |x(2)| = |(17, 0, 5)| = , and |v(2)| = |(12, -1, 2)| =
• At t = - 2 , |x(- 2)| = |(17, -12, -3)| = , and |v(- 2)| = |(- 12, 7, 2)| =

Notice that the magnitude of the creature's velocity (i.e. the speed at which the creature is traveling) is high at t = - 2 , decreases considerably at t = 0 , and goes back up again at t = 2 , even though the acceleration is constant! This is because the acceleration is causing the creature to slow down and change direction--in the same way that a ball thrown upwards (which experiences constant acceleration due to the earth's gravity) slows down to zero-velocity as it reaches its maximum height, and then changes direction to fall back down.