1. Understanding Motion
Linear Motion

2. Motion
The motion of an object can only be recognized when something is established as a basis of comparison…a reference point
We say an object is moving when its position changes compared to that reference point
For most day-to-day situations, the Earth, and those things affixed to it, serve as a convenient reference frame.

3. Position and Time
These are the two most fundamental physical quantities that can be measured to describe an object’s motion.
The relationship between these variables can be discovered experimentally and modeled using mathematics in both graphical and equation form.

4. Position vs. Time Graphs
The magnitude (size) of the slope tells us…
The algebraic sign of the slope tells us…
The magnitude and sign together tell us…
The vertical intercept tells us…
When this graph is a straight line we know…

5. Position vs. Time Graphs
The generic equation for a linear graph is… y = mx + b In terms of the physical quantities being plotted this becomes… x = mt + b
Position, x (m)
Time, t (s)

6. Position vs. Time Graphs
If we replace the slope and intercept terms with what they tell us we get… x = vt + xo Where v is the velocity (m/s) and xo is the initial position (m) of the object in motion
m = v
Position, x (m) xi
Time, t (s)

7. x = vt + xo
This equation (straight line with slope v and intercept xo) is a model that describes the relationship between position and time for an object moving with constant velocity.
x = position of object after time, t
v = velocity of object (speed in a direction)
t = elapsed time
xo = starting position of the object

8. Ball on a ramp questions:
What information does the slope of an x-t graph tell us?
Does your x-t graph for the ball on a ramp have slope?
What does the shape of your x-t graph tell us about the motion of the ball?
Does the information from your v-t graph support the description given in #3? How so?
What information does the slope of a v-t graph tell us?
What information does the intercept of the v-t graph tell us?
What value would you expect for the intercept in this lab activity? Why?

9. Constantly Accelerated Motion (ball on a ramp)
Velocity-time graphs are linear… v  t
Slope is constant  The rate at which velocity is changing is constant SLOPE = ACCELERATION
Position-time graphs are NOT linear, they are quadratic… x  t2
slope is NOT constant  Velocity is changing

10. vf = at + vo
This equation is a model describing the relationship between velocity and time for an object that is constantly accelerating
vf – “final” velocity after time, t
a – acceleration (slope of v-t graph)
t – elapsed time
vo – starting velocity (intercept of v-t graph)

11. x = ½ at2 + vot + xo
This equation models the relationship between position and time for constantly accelerated motion
This equation emerges from our ball on a ramp data or it could be derived (we will!)
x = position after time, t
a = acceleration
vo = starting velocity
t = elapsed time
xo = starting position

12. A summary of motion equations thus far…
Constant velocity: x = vt + xo  x = vt or v= x/t (v is constant or average velocity) Constant acceleration: vf = at + vo  a = v/t x = ½ at2 + vot + xo  x = ½ at2 + vot