x = 0 m sometimes (in fact most of the time) objects do not move at a constant velocity. for...
TRANSCRIPT
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.
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
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
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
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.
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.
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
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.
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,
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.
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
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.