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Instantaneous Velocity
So we have found a way to assign a number to describe the motion over an interval of time, but we
have not been able to describe the details of the motion.
We can improve the detail of our description of the motion by making the time interval shorter.
The extreme, or limit, of making the time interval shorter is called a instant.
The velocity over an arbitrarily short interval, or instant, is called the instantaneous velocity.
๎ v = lim
๎ญ t ๎ 0
๎ญ ๎ x
๎ญ t
The invention of the calculus supplied the mathematical rigor to the above definition, and the final
solution to the problem of the measurement of motion.
Although we are not going to learn calculus in this course we can make a see what it does by
examining the below graph of an object position versus time.
In graph (a.) we see that a line which connects 2 points on the position curve. The slope of this line is
the rise over the run and it is
slope =
๎ญ ๎ x
๎ญ t
which is just our definition of the average velocity over the interval ๎ญ t
In the second graph we see that there is a line which is tangent to the curve, and therefore touches the
curve at only one point. The progress from graph a to graph b is a graphical representation of taking
the limit as ๎ญ t goes to zero. The 2 point which define the interval move closer and closer together
till they become one.
The instantaneous velocity is the slope of the straight line which is tangent to the position vs time
curve.
Constant Velocity
If we have a make several measurements of the instantaneous velocity of an object over a time interval,
and these measurement do not change then we say that the object is moving with a constant velocity
over that time interval.
Speed and Velocity
Velocity is a vector which depends on position which is also a vector.
If we apply the same scheme for distance , which is a scalar, we get the concept of average and
instantaneous speed.
Speed is the ratio of the change in distance over the change in time. If I calculate the distance between
an object and some starting point s , then the speed will be given by
v = lim
๎ญ t ๎ 0
๎ญ s
๎ญ t
Note that there is no little arrow above the v indicating that it is a scalar.
The magnitude of the velocity vector is the speed.
v =
v
Changing velocity --- acceleration
Since we have defined instantaneous velocity we can now talk about the change in velocity.
We can form a mathematical object strictly analogous to the definition of velocity, the ration of the
change in velocity to the time interval.
a
av
๎ญ ๎ v
๎ญ t
v
2
v
1
t
2
โ t
1
Furthermore we can define instantaneous acceleration
๎ a = lim
๎ญ t ๎ 0
๎ญ ๎ v
๎ญ t
If we consider an analogy to the position vs time graph we conclude that
The acceleration is the slope of the tangent to the velocity vs. time graph.
Let's try these formulae with some examples.
Ex. A car is found to be traveling in the positive direction at a velocity of 10 miles per hour, at time =
5 seconds. At time = 9 seconds the car have come to a stop. Calculate the average acceleration of the
car for the interval from 5 to 9 seconds.
To solve the problem we first write down the mathematical relationship we are going to use.
a
av
โ 4
miles
sec
sec
This means that the velocity changes by โ6.94โ 10
โ 4
miles
sec
every second.
A special case: Constant Acceleration
Equations of Constant Acceleration.
If we state algebraically the above conclusions we get the following 4 equations for an object which
starts out at moving at a velocity
v
0
at time t = 0 , and accelerates at a constant acceleration a
v = ๎
v
0
a t (2-11a)
๎ x = x ๎
0
๎ v ๎
0
t ๎
๎ a t
2
(2-11b)
v
2
= v
0
2
๎2a๎ x โ x
0
(2-11c)
In addition we have, for cases of constant acceleration
v ๎
av
v ๎ ๎
v
0
(2-11d)
We see from Equation 2-11b that
For an object moving with constant acceleration the position is proportional to the SQUARE OF
THE TIME
Galileo's Experiment
Galileo determined that object which we rolling down a smooth ramp, under the influence of gravity,
the distance was precisely proportional to the elapsed time SQUARE.
The reverse inference is also valid, object whose displacement is proportional to the square of the time
are undergoing constant acceleration.
Homework
Ch. 2. Problem #6, #10 , #17, #25 (Use Student Solution manual to find similar worked out
problems)
CLICKER QUIZ
A car slows down from 23 m/s to rest in a distance of 85 m. What is its acceleration?
