1. INSTANTANEOUS speed and velocity on x-t graphs

2. A quick review * On an x-t graph, when you “RISE” up on the graph, your position (x) changes. * On an x-t graph, when you “RUN” to the right on a graph, your time “t” changes. * Since avg velocity is Dx/Dt, then the RISE/RUN of an x-t graph is …. avg velocity.

3. Equation is for Honors Only!!!
Definition
Instantaneous Velocity (v) – the velocity of an object at a precise moment in time.
v = lim(Dx/Dt)
Dt0
Equation is for Honors Only!!!

4. Just what is meant by “instantaneous” velocity?Dt Dt Dt Dt
Finally, “in the limit” that the time interval is infinitely small (or approximately zero), we find the velocity at a single moment in time.
 Hence the term
“instantaneous velocity” Dt
To find the average velocity between two points in time, we find the slope of the line connecting these two points, thus finding the change in position (rise) over the change in time (run).
As the two points move closer together, we find the average velocity for a smaller time interval.
As the two points move EVEN CLOSER together, we find the average velocity for an EVEN SMALLER time interval.

5. To find instantaneous velocities, we still use the concept of slope
To find instantaneous velocities, we still use the concept of slope. But we find the slope OF THE TANGENT TO THE CURVE at the time in question
Definition
Tangent to a Curve – a line that intersects a given curve at exactly one point.

6. x t The slope of the tangent tells you about the object’s velocity.
The more “slopey” the graph, the faster the object moves

7. Good Tangents 
Bad Tangents 

8. How to find the instantaneous velocity of a specific time interval from an x-t graph …
x(m)
t (s) 30 20 10
Example:
What is the instantaneous velocity at t = 20 seconds?
(24, 30)
(15, 15)
Draw the tangent to the curve at the point in question. Then, find the slope of the tangent.
Slope = rise/run = 15 m / 9 s
= 1.7 m/s (approx)
YOU MUST SPECIFY WHICH POINTS YOU USED WHEN FINDING THE SLOPE!!!!

9. How to find the instantaneous velocity of a specific time interval from an x-t graph …
x(m)
t (s) 30 20 10
Example:
What is the instantaneous velocity at t = 5?
If the pt lies on a segment, find the slope of the segment.
Slope = 5 m / 10 s = 0.5 m/s
(0,5)
(10,10)
YOU MUST SPECIFY WHICH POINTS YOU USED WHEN FINDING THE SLOPE!!!!

10. How to find the instantaneous velocity of a specific time interval from an x-t graph …
x(m)
t (s) 30 20 10
Example:
What is the instantaneous velocity at t = 25 seconds?
Draw the tangent to the curve at the point in question. Then, find the slope of the tangent.
Slope = 0 (object at rest)

11. x-t graphs 2 1 3 x (m) x2 x1 x3 t (sec) t1 t2 t3 Slope of line segment
Slope of line segment
Slope of line segment

12. (13.5,-20)
(0,6)
(33,2)
(11,-20)
Tangent to the curve has a slope of +22m / 22sec = 1
Speed is 1 m/s (no sign, scalar), Velocity is +1 m/s (needs sign, vector).
Tangent to the curve has a slope of -26m / 13.5s = m/s
THEREFORE, v = m/s and s = 1.93 m/s (approximately)