Graphing Motion

Equations are not the only way to go.

We can often use graphs to quickly get a sense of what is happening as an object moves:

• Where is the object, and when.

• How fast is it moving and in what direction.

• By how much did it change it’s position.

• What’s the object’s acceleration.

Making and analyzing graphs is a cool skill, necessary for every burgeoning junior physicist.

Consider the following data for a student walking to school.

Plot the points on a graph with position (x) on the vertical axis, and time (t) on the horizontal axis.

This is a position-vs-time graph, also known as a position graph, or an x-vs-t graph.

Important! The graph does not show the object’s trajectory. Remember, the object is moving along the x- axis.

x (m)

t (s)

1. Where did the object start?2. In what direction did the object start moving?3. At what time did the object cross the origin?4. When was the object 5 m left of the origin?5. How fast was the object moving at t = 35 s?6. What total distance did the object cover?7. What was the object’s average speed for the whole

trip?8. What was the object’s total displacement?9. What was the object’s average velocity for the whole

trip?

What was the object’s velocity for 0 < t < 2 s?What was the object’s velocity for 2 < t < 4 s?What was the object’s velocity for 4 < t < 6 s?

To go from a position graph to a velocity graph:

Find the slopes…

Then plot the slope values.

Constant Acceleration

On a position graph, constant acceleration is always a parabola.

Average velocity for any time interval is the slope of the secant line over that

time.

Finding vavg from t = .75 s to t = 1.5 s

Secant line

What if we want instantaneous velocity?

Instantaneous velocity at a particular time is the slope of the tangent line at

that time.

Finding v at t = .75 s

Tangent line

Tangent line

Secant line

Slope = average velocity

Slope = instantaneous velocity

Displacement is the area under the velocity-vs-time graph.

Acceleration is the slope of the velocity graph.

c

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