Question

You are operating a remote-controlled model car on a vacant tennis court. Your position is the origin of coordinates, and the surface of the court lies in the 𝑥𝑦-plane. The car, which we represent as a point, has x- and y- coordinates that vary with time according to

 a) Find the car’s coordinates and its distance from you at time t = 2.0 s.

b) Find the car’s displacement and average velocity vectors during the interval from

t = 0 s to t = 2.0 s.

c) Find the components of the average acceleration in the interval from t = 0 s to t =

2.0 s.

Answer 1

Explanations & Calculations

• The question is incomplete without the relation of how X and Y coordinates are connected.
• But you may have it and it should look like $\small X_{(t)}\,\,\&\,\,Y_{(t)}$

a)

• Just substitute t = 2.0 into x in both $\small X(t)\,\&\quad Y(t)$ to get the coordinates after time being 2.0s.
• Then perform the usual Pythagorean distance calculation to find the distance $\small L=\sqrt{X_{(t)}^2+Y_{(t)}^2}$

b)

• Now write a vector relation with the unit vectors associated with the x and y axes for the 2 points it may be at t=0 and t=2 ; $\small \vec{r}=X_{(t=2)}\hat i+Y_{(t=2)}\hat j$
• Get the first derivative of this vector function with respect to time, to find the average velocity vector.

c)

• Finally, differentiate the average velocity vector once again w.r.t time and get the acceleration vector.
• Then to get the x and y components, just consider the coefficients of unit vectors i and j.