Average Velocity and
Instantaneous Velocity
The speedometer of the car indicates how fast the
car is moving at a given period of time.
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In an analog
speedometer if the car
moves at a fast rate,
the pointer will point
towards the higher
number stating that the
rate of change of
position of the car is
higher at that specific
time.
An analog speedometer displaying
a reading in unit of kilometers per
hour (kph)
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In real life, objects usually do not travel at a
constant speed. In this lesson, we will be
describing how we can determine the
instantaneous velocity of an object.
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How do we describe the
instantaneous speed (or velocity)
of an object?
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How can we differentiate average
velocity from instantaneous
velocity?
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Velocity
Average Velocity
It is defined as the total rate of change of position of an object.
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Velocity
Average Velocity
The equation for average velocity can be used to determine
which would reach the final point in the shortest amount of
time.
10 kph
15 kph
20 kph
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Velocity
Average Velocity
However, it is not enough to tell us about how fast and in what
direction an object is moving at any particular time.
Arf arf!
35 kph
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Velocity
Instantaneous Velocity
It is defined as the velocity of an object in an instant. It means
that we are getting the rate of change in the position of an
object in a single point of time.
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Velocity
Instantaneous Velocity
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Velocity
Instantaneous Velocity
The velocity is the slope in a
position-time graph.
The preciseness increases
with smaller time interval.
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Velocity
Instantaneous Velocity
In calculus, the instantaneous
velocity can be solved by getting
the derivative of the position-time
equation.
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A negative value of vins means that x is
decreasing and the motion is in the
negative x-direction. A body can have
positive x and negative vins, or the
reverse. The position (x) only tells us
where the object is relative to the origin
and the vins tells us the direction of the
movement of the object.
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The position of a particle is given by the equation
below. Determine the instantaneous velocity of the
particle at t = 2s.
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The position of a particle is given by the equation
below. Determine the instantaneous velocity of the
particle at t = 2s.
Assuming all quantities are expressed in SI units, at t = 2
s, the particle is moving at 26.8 m/s.
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Calculate for the instantaneous velocity of
a car with a position as a function of time
equation of x(t) = 5t3 + 6t4 + 5 at
t=
0.25 s. Assume that all quantities are
expressed in SI units.
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Juan ran as fast as he can in front of a motion sensor.
The motion sensor recorded his x(t) equation as shown
below. How much did his velocity change from t = 1 s to
t = 3 s? Did he slow down or speed up?
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Juan ran as fast as he can in front of a motion sensor.
The motion sensor recorded his x(t) equation as shown
below. How much did his velocity change from t = 1 s to
t = 3 s? Did he slow down or speed up?
The difference between the instantaneous velocities is
‒49.08 m/s. The negative sign suggests that Juan is
slowing down.
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A motion sensor had an output equation
for a flying bird of
x(t) = 2.4t3 +
4.2 + 5.5t2. What is the difference between
the instantaneous velocities of the bird at
t = 1 and t = 3? Evaluate your answer.
Assume that all quantities are expressed
in SI units.
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A hypothetical car stops at an intersection. When the
stop light turns green, it then travels along a straight
road and its distance from the intersection is given by
x(t) = at2 + bt3 – c where a = 4.6 m/s2, b = 1.2 m/s3 and c
= 17 m. Calculate for the average velocity of the jeep
from t = 2 s to t = 8 s and its instantaneous velocity at t
= 9 s. Do you think this hypothetical car can be usually
seen in our streets?
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Do you think this hypothetical car can be usually seen
in our streets?
The average velocity of the car from t = 2 s to t = 8 s is
146.8 m/s and its instantaneous velocity at t = 9 s is
374.4 m/s. This car is just hypothetical because if we
convert the values into km/h, 146.8 m/s is approximately
equal to 530 km/h and 374.4 m/s is 1347 km/h. We do not
see cars of these velocities in our streets.
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A particle has a position as a function of
time equation of x(t) = 2.3t3 + 2t2 + 4t – 2.4.
Calculate for the average velocity of the
particle from t = 4 s to t = 6 s and its
instantaneous velocity at t = 3 s.
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Why do we use differential
calculus in solving for the
instantaneous velocity of a
particle?
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A. Write T if the statement is correct and F if
not.
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1. Any object traveling along a straight has a constant velocity.
2. Average velocity is synonymous with instantaneous velocity.
3. The slope of a position vs. time graph is velocity.
4. If the average velocity is high, the instantaneous velocity is also
high.
5. Instantaneous velocity is the velocity at an instant.
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6. The average velocity only considers the total displacement and
total time elapsed.
7. One of the key differences between average velocity and
instantaneous velocity is that the latter is a scalar quantity.
8. The values of the instantaneous velocity can tell us that an object
is either speeding up or slowing down.
9. The derivative of the velocity as a function of time equation gives
you instantaneous velocity.
10. Instantaneous velocity can never be equal to the average
velocity.
ANSWERS:
1.
2.
3.
4.
5.
6.
7.
8.
9.
False
False
True
False
True
True
False
True
True
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Complete the table below. Assume that the equations in
the first column have homologous units.
Equation
Time (s)
Instantaneous
Velocity (m/s)
t=2
t=9
t=4
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● Average velocity is defined as the total rate of
change of position of an object.
● Instantaneous velocity is the velocity of an object in
an instant.
● When an object is traveling along a straight line, the
velocity and speed of an object are equal.
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● The slope of a position vs. time graph is velocity.
● In order to calculate the instantaneous velocity of an
object, we have to get a small segment of
displacement (Δx) and divide it with its infinitesimal
time (Δt). This is the same as getting the first
derivative of the position as a function of time
equation.
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Concept
Formula
Average Velocity
where
●
is the
average velocity
●
●
is displacement
t is time
Description
Use this equation to
calculate the average
velocity if the total
displacement and the
time elapsed are both
given.
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Concept
Formula
Instantaneous
Velocity
where
●
is the
instantaneous velocity
●
is the change in
position
is time elapsed
●
Description
Use these equations to
get the instantaneous
velocity of a particle
with a non-constant
velocity. Both of these
equations are
synonymous and
calculus-based.
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Concept
Formula
Instantaneous
Velocity
where
●
is the
instantaneous velocity
●
is the derivative
of the x(t) equation
Description
Use these equations to
get the instantaneous
velocity of a particle
with a non-constant
velocity. Both of these
equations are
synonymous and
calculus-based.
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PROBLEM 1
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• A car has travelled a distance of 30m in 10 seconds from point A to
point B. (a) What is the average speed of the car? (b) If the
displacement of the car is 20 meters, north, what is the car’s
average velocity?
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ACCELERATION
• A vector quantity which describes how velocity changes with time.
• It can also be expressed in units such as:
• m/s²
• Km/hr²
• Km/h/s
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A moving object is accelerating when”
1. The velocity is changing (increasing or decreasing)
2. the direction is changing
3. both velocity and direction is changing.
Acceleration can be positive or negative.
Deceleration indicates negative acceleration.
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Uniformly accelerated motion has the following
features:
1. Distance traveled after each unit of time are
not equal.
2. The graph of distance against time is a
parabola; hence distance is directly proportional
to time squared.
3. The graph of speed against time is straight line.
The slope of a velocity vs time graph represents
constant acceleration.
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• Not all accelerations are constant. But in motion
can best understood in terms of acceleration. A
freely-falling body is an example of motion of
uniform acceleration.
PROBLEM 1
• As a shuttle bus comes to a normal stop, it slows from 9.0 m/s to
0.0m/s in 5.0s. Find the average acceleration of the bus.
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PROBLEM 2
• During a race, a sprinter increases from 5.0 m/s to 7.5 m/s over a
period of 1.25s. What is the sprinter’s average acceleration during
this period?
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PROBLEM 1
• A driver brings a car travelling at 20 m/s to full
stop in 2 sec.
• (A) What is the car’s acceleration?
• (b) How far does it travel before stopping?
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PROBLEM 2
• What is the acceleration of a car that goes from
20mi/hr to 20 mi/hr in 1.5 seconds?
• At the same acceleration, how long will it take
for the car to go from 30 mi/hr to 36 mi/hr?
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