Calculus Vectors

Question

The position of a particle in the​ xy-plane at time t is r(t)=(t+4) i+(t^2−2)j.

Find an equation in x and y whose graph is the path of the particle. Then find the​ particle's velocity and acceleration vectors at t=1.

Answer 1

There is a fun way to see how to write y in terms of x.

Note how the parametrization consists of shifting x 4 to the right of t.

Thus, we may write y = (x-4)^2 -2

The particles velocity is (1, 2t)

The acceleration is (0, 2)

Thus, v(1) = (1,2*1) = (1,2) and ||v|| = √5

a(1) = (0, 2), so ||a|| = 2