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![Page 1: x = 0 m Sometimes (in fact most of the time) objects do NOT move at a constant velocity. For example, consider the ball starting at the point x =](https://reader036.fdocuments.in/reader036/viewer/2022062722/56649f385503460f94c544c3/html5/thumbnails/1.jpg)
x = 0 m
Sometimes (in fact most of the time) objects do NOT move at a constant velocity.For example, consider the ball starting at the point x = 0 m when t = 0 s.
Topic 2.1 ExtendedA – Instantaneous velocity
x(m)
t = 0
st =
1 s
t = 2
s
t = 3
s
t = 4
sx = 2 m
x = 8 m
x = 18 m
x = 32 m
The ball has the smallest velocity at the beginning of its motion, and the biggest velocity at the end of its motion.
![Page 2: x = 0 m Sometimes (in fact most of the time) objects do NOT move at a constant velocity. For example, consider the ball starting at the point x =](https://reader036.fdocuments.in/reader036/viewer/2022062722/56649f385503460f94c544c3/html5/thumbnails/2.jpg)
x = 0 m
To get a handle on its motion, we graph the data.We begin by making a table of values.
Topic 2.1 ExtendedA – Instantaneous velocity
x(m)
t = 0
st =
1 s
t = 2
s
t = 3
s
t = 4
sx = 2 m
x = 8 m
x = 18 m
x = 32 m
t(s) x(m)
0 0
1 2
2 8
3 18
4 32
![Page 3: x = 0 m Sometimes (in fact most of the time) objects do NOT move at a constant velocity. For example, consider the ball starting at the point x =](https://reader036.fdocuments.in/reader036/viewer/2022062722/56649f385503460f94c544c3/html5/thumbnails/3.jpg)
Then we plot the data on a suitable coordinate system:
Topic 2.1 ExtendedA – Instantaneous velocity
0
x(m
)
t(s)1 2 3 4
32
18
8
20
t(s) x(m)
0 0
1 2
2 8
3 18
4 32
![Page 4: x = 0 m Sometimes (in fact most of the time) objects do NOT move at a constant velocity. For example, consider the ball starting at the point x =](https://reader036.fdocuments.in/reader036/viewer/2022062722/56649f385503460f94c544c3/html5/thumbnails/4.jpg)
Note that the average velocity now depends on WHICH TWO POINTS WE CHOOSE:
Topic 2.1 ExtendedA – Instantaneous velocity
0
x(m
)
t(s)1 2 3 4
32
18
8
20
Which slope is the best estimate for the velocity at t = 0.5 seconds?
Which slope is the best estimate for the velocity at t = 2.5 seconds?
FYI: The closer together the two points on the graph are, the better the estimate of the velocity of the particle BETWEEN THE TWO POINTS.
V0.5
V2.5
![Page 5: x = 0 m Sometimes (in fact most of the time) objects do NOT move at a constant velocity. For example, consider the ball starting at the point x =](https://reader036.fdocuments.in/reader036/viewer/2022062722/56649f385503460f94c544c3/html5/thumbnails/5.jpg)
Recall that the average velocity was given by
Topic 2.1 ExtendedA – Instantaneous velocity
0
x(m
)
t(s)1 2 3 4
32
18
8
20
v = xt
Average Velocity
We now define the instantaneous velocity v like this:
v =limitt→0
xt
Instantaneous Velocity
The above is read "vee equals the limit, as delta tee approaches zero, of delta ex over delta tee."
FYI: In words, the instantaneous velocity is the velocity of the MOMENT, not some sort of average.
FYI: The key is to choose your times VERY CLOSE TOGETHER to get the most exact value for v.
FYI: That is the meaning of t→0.
![Page 6: x = 0 m Sometimes (in fact most of the time) objects do NOT move at a constant velocity. For example, consider the ball starting at the point x =](https://reader036.fdocuments.in/reader036/viewer/2022062722/56649f385503460f94c544c3/html5/thumbnails/6.jpg)
Newton understood that CHANGE was a characteristic of the physical world.
Topic 2.1 ExtendedA – Instantaneous velocity
0
x(m
)
t(s)1 2 3 4
32
18
8
20
Most objects, like the planets, cars, atoms, and the ball whose motion we are analyzing, exhibit changing motion.Newton understood that a new branch of mathematics, called calculus, would be necessary to gain an in-depth understanding of changing motion.
Thus, in 1665, in conjunction with his development of physics, Newton invented calculus - the study of change.
FYI: All branches of the so-called "hard" sciences use calculus. The fields of engineering, statistical analysis, computer science, and even political science, use calculus. The financial industry, insurance industry and businesses all use calculus...
FYI: The reason is simple: Everything man studies or influences exhibits change. Calculus is the STUDY of CHANGE.
![Page 7: x = 0 m Sometimes (in fact most of the time) objects do NOT move at a constant velocity. For example, consider the ball starting at the point x =](https://reader036.fdocuments.in/reader036/viewer/2022062722/56649f385503460f94c544c3/html5/thumbnails/7.jpg)
In fact, calculus looks at INFINITESIMAL (small) changes, and finds their CUMULATIVE EFFECTS.
Topic 2.1 ExtendedA – Instantaneous velocity
0
x(m
)
t(s)1 2 3 4
32
18
8
20
The smaller the change, the more exact the predictions made by the analysis.
v =limitt→0
xt
Thus, instantaneous velocity
is more useful than average velocity
v = xt
![Page 8: x = 0 m Sometimes (in fact most of the time) objects do NOT move at a constant velocity. For example, consider the ball starting at the point x =](https://reader036.fdocuments.in/reader036/viewer/2022062722/56649f385503460f94c544c3/html5/thumbnails/8.jpg)
As an illustration, suppose we want to know the actual speed of the ball at t = 2 s.
Topic 2.1 ExtendedA – Instantaneous velocity
0
x(m
)
t(s)1 2 3 4
32
18
8
20
As my first approximation, I choose t = 2 s and t = 4 s as my two data points:
v = xt
32 - 84 - 2
= = +12 m/s
t
x As my second approximation, I choose t = 2 s and t = 3 s as my two data points:
v = xt
18 - 83 - 2
= = +10 m/s
t
x
The second approximation for the velocity at t = 2 s is better. Why?
FYI: Observe that the smaller t is, the better the approximation.
![Page 9: x = 0 m Sometimes (in fact most of the time) objects do NOT move at a constant velocity. For example, consider the ball starting at the point x =](https://reader036.fdocuments.in/reader036/viewer/2022062722/56649f385503460f94c544c3/html5/thumbnails/9.jpg)
As an illustration, suppose we want to know the actual speed the ball at t = 2 s.
Topic 2.1 ExtendedA – Instantaneous velocity
0
x(m
)
t(s)1 2 3 4
32
18
8
20
Without an accurate graph, on very fine graph paper, it is difficult to get much better values for the speed at t = 2 s (which we now have estimated at 10 m/s).
t
x
t
x
In order to keep on going, we need an analytic form for the data. What this means is that we need a formula.Without going into detail, it turns out that the formula for x is given by since this formula exactly replicates the data:
t x = 2t2
0 x = 2·02 = 0
1 x = 2·12 = 2
2 x = 2·22 = 8
3 x = 2·32 = 18
4 x = 2·42 = 32
x = 2t2,
![Page 10: x = 0 m Sometimes (in fact most of the time) objects do NOT move at a constant velocity. For example, consider the ball starting at the point x =](https://reader036.fdocuments.in/reader036/viewer/2022062722/56649f385503460f94c544c3/html5/thumbnails/10.jpg)
As an illustration, suppose we want to know the actual speed the ball at t = 2 s.
Topic 2.1 ExtendedA – Instantaneous velocity
0
x(m
)
t(s)1 2 3 4
32
18
8
20
Now we can move our second point as close to t = 2 s as we want.
x = 2t2
For example, at t = 2.5 seconds x = 2(2.5)2 = 12.5 m so that
v = xt
12.5 - 82.5 - 2
= = +9 m/s
FYI: We may make our second time as close to our first as we want: The closer it is, the more accurate the velocity.
![Page 11: x = 0 m Sometimes (in fact most of the time) objects do NOT move at a constant velocity. For example, consider the ball starting at the point x =](https://reader036.fdocuments.in/reader036/viewer/2022062722/56649f385503460f94c544c3/html5/thumbnails/11.jpg)
As an illustration, suppose we want to know the actual speed the ball at t = 2 s.
Topic 2.1 ExtendedA – Instantaneous velocity
Observe the average velocity as the second point approaches the first (in this case t = 1 s):
0
x(m
)
t(s)1 2 3 4
32
18
8
20
x = 2t2
FYI: Note that the closer the second point is to the first, the closer the slope is to that of the TANGENT line.
FYI: In fact, the INSTANTANEOUS VELOCITY at a point on the graph of x vs. t is equal to the SLOPE OF THE TANGENT at that point.
TANG
ENT
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Suppose the x vs. t graph of a particle looks like this:
Topic 2.1 ExtendedA – Instantaneous velocity
(a) Sketch in the instantaneous velocity tangents for various points on the curve:
x
t
FYI: In fact, the INSTANTANEOUS VELOCITY at a point on the graph of x vs. t is equal to the SLOPE OF THE TANGENT at that point.
tA tB
--
-
-
(b) Label the slopes that are zero with a 0:
0
0
(c) Label the slopes that are negative with a -:
(d) Label the slopes that are positive with a +:
+
(e) Label the time where the particle first reverses direction tA:(f) Label the time where the particle next reverses direction tB:
FYI: The INSTANTANEOUS VELOCITY is zero where a particle REVERSES direction.