Question

1. A car is 3.0 km west of a traffic light at t = 0 and 7.0 km east of the light at t = 8.0 min. Assume the origin of the coordinate system is the light and the positive x direction is eastward. (a) What are the car's position vectors at these two times? (b) What is the car's displacement between 0 min and 6.0 min?
2.  The position of a particle moving along the x-axis is given by x(t) = 5.0 − 2.0t m. (a) At what time does the particle cross the origin? (b) What is the displacement of the particle between t = 4.0 s and t = 8.0 s?
3. A particle moves along the x-axis according to x(t) = 8t - 2t² m. (a) What is the instantaneous velocity at t = 2 s and t = 3 s? (b) What is the instantaneous speed at these times? (c) What is the average velocity between t = 2 s and t = 3 s?

Answer 1

1 (a) The position vectors:

$$r_1=-3i,\\ r_2=+7i.$$

1 (b) The displacement in 8 minutes:

$$\Delta x_8=7-(-3)=10\text{ km}.$$

This took 8 minutes, so, the velocity is

$$v=\frac{\Delta x_8}{t_8}=5/4\text{ km/min}.$$

Assuming constant velocity, at t=6 min we have

$$\Delta x_6=vt_6=\frac54·6=7.5\text{ km}.$$

2 (a) The particle passes the origin when the origin is 0:

$$x(t) =0= 5.0 − 2.0t,\\ t=2.5\text{ s}.$$

2 (b) The displacement between 4 and 8 s:

$$\Delta x=x(8)-x(4),\\ \Delta x=[5-2·8]-[5-2·4]=-8\text{ m},$$

which means that at 8 s the car is further from the origin than at 4 s.

3 (a) The velocities are the derivatives of displacement over time:

$$v(t)=x'(t)=8-4t,\\ v(2)=0,\\ v(3)=-4\text{ m/s}.$$

3 (b) The instantaneous speeds are equal to the instantaneous velocities in magnitude:

$$s(2)=0,\\ s(3)=4\text{ m/s}.$$

3 (c) The average velocity is displacement over time (1 s):

$$x(2)=8·2-2·2^2=8\text{ m},\\ x(3)=6\text{ m},\\ \Delta x=-2\text{ m},\\ v_{av}=\frac{\Delta x}{t}=-2\text{ m/s}.$$