Warm-Up 4/8 Give equation for each graph. y = 2sin x y = sin x Q&A on assignment.
INSTANTANEOUS speed and velocity on x-t graphs. A quick review * On an x-t graph, when you...
12
INSTANTANEOUS speed and velocity on x-t graphs
Transcript of INSTANTANEOUS speed and velocity on x-t graphs. A quick review * On an x-t graph, when you...
INSTANTANEOUS speed and velocity on x-t graphs
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 x/t, then the RISE/RUN of an x-t graph is ….
avg velocity.
DefinitionInstantaneous Velocity (v) – the velocity of
an object at a precise moment in time.
v = lim(x/t)t0 Equation is for
Honors Only!!!
Just what is meant by “instantaneous” velocity?
t
ttt
t
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.
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”
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.
x
t
The slope of the tangent tells you about the object’s velocity.
The more “slopey” the graph, the faster the object moves
Good Tangents
Bad Tangents
How to find the instantaneous velocity of a specific time interval from an x-t graph …
x(m)
10 20 30 40 50
t (s)
30
20
10
0
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)
Example:
What is the instantaneous velocity at t = 20 seconds?
YOU MUST SPECIFY WHICH POINTS YOU USED WHEN FINDING THE SLOPE!!!!
(24, 30)
(15, 15)
How to find the instantaneous velocity of a specific time interval from an x-t graph …
x(m)
10 20 30 40 50
t (s)
30
20
10
0
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!!!!
How to find the instantaneous velocity of a specific time interval from an x-t graph …
x(m)
10 20 30 40 50
t (s)
30
20
10
0
Draw the tangent to the curve at the point in question. Then, find the slope of the tangent.
Slope = 0 (object at rest)
Example:
What is the instantaneous velocity at t = 25 seconds?
x-t graphs
t (sec)
x (m)
t1 t2 t3
x2
x1
x3
1
2
3
0
01
011to0 tt
xxv
Slope of line segment
02
022to0 tt
xxv
Slope of line segment
linetangent
ofslopev 3ptat
Tangent to the curve has a slope of -26m / 13.5s = -1.93 m/s THEREFORE, v = -1.93 m/s and s = 1.93 m/s
(approximately)
Tangent to the curve has a slope of +22m / 22sec = 1Speed is 1 m/s (no sign, scalar), Velocity is +1 m/s (needs sign, vector).
(13.5,-20)
(0,6) (33,2)
(11,-20)
