# Week 4 (Sep. 2 - 6)

**Reading:**PSE Chap 3, Vectors

**Topics:**This week, we continue our discussion of week 3 on one-dimensional kinematics and the mean speed theorem. I will introduce the equations of one-dimensional kinematics, and will introduce

*vectors*and

*vector algebra*

**No quiz this week.**

**Week 4 Homework Problems and Ramp lab:**

**Ramp laboratory experiment**

- Set up a ramp. Measure the angle of the ramp with respect to the horizontal desktop.
- Roll a small steel ball down the ramp. Record the time the ball takes to roll 10, 20, 30, 40,
*etc.*cm down the ramp. Be sure to record experimental uncertainty. - Repeat this procedure for three different ramp angles.
- Make a plot of the distance (ordinate) as a function of time (abscissa) using graphical analysis software. Label your plot appropriately.
- Put the data from all three data sets on the same graph; for each data set perform a power-law fit to the data. Does your fit match your expectations?
- Using the kinematic equations we've learned this week, determine the acceleration of the ball from your graphs.

**Accelerating car:**A car, initially at rest travels 20 meters in 4 seconds along a straight line with constant acceleration. What is its acceleration? Make a plot of the position, velocity and acceleration of this car. Solution.

**Accelerating truck:**How far does a truck travel in 6 seconds if its initial velocity is 2 m/s and its acceleration is 2 m/s^2 in the forward direction? Solution.

**Accelerating particle:**A particle's position is given by x(t) = 12 t - 3.0 t^2. What is v(t)? What is a(t)? Make a sketch of its position, velocity and acceleration versus time. Is the particle ever at rest? If so, when? Note: this problem uses

*derivatives,*a method of calculus with which some of you may not (yet) be familiar. Solution.

**Classroom exercises:**As a reminder, these are some of the problems we worked out in class…

- Finding average velocities from displacement vs. time plots
- Finding the time of flight and height of a ball thrown upwards.
- 1-d kinematic equations: the connection between geometry (area under velocity vs time plots), algebra (kinematic equations relating position, velocity and acceleration), and calculus (integrating acceleration to find velocity and again to find position).