Physics 30S - Position Time Graphs: Types of Velocities

Physics 30S - Position – Time Graphs: Types of Velocities

There are 3 types of velocities that we will consider. 1. Constant Velocity 2. Average Velocity 3. Instantaneous Velocity

1. Constant Velocity

You are driving down a highway and you notice that it takes equal intervals of time to travel between hydro poles. This is an indication that you are traveling at constant velocity. The formula for determining constant velocity is:

d v - velocity in m/s, km/h, etc. v  t d - displacement in m, km, etc. t - time intervals in s, or h

Note that since displacement is either a positive or negative value, velocity will have the same directional notation. The similarity in sign for both velocity and displacement occurs because time interval is always positive.

Practice

Note that when writing the direction of a measurement we can use the actual direction. It is in a math formula that we need to substitute "+" and "-" for direction notation.

1) What is the constant velocity of an airplane that flies 602 m E in 2.5 s? Express your answer in meters per second and in kilometers per hour.

2) The tine on a tuning fork moves 1.0 mm to the right in 4.0 x 10-3s. What is the constant velocity of the tine in meters per second?

- 1 - Physics 30S - Position – Time Graphs: Types of Velocities 3) An electron travels at a constant velocity of 1.3 x 105 m/s. How much time is required for a displacement of 1.00 m?

4) A spaceship traveled at a uniform velocity of 3.2 x 104 km/h for 2.7 days. What was the displacement of the spaceship?

5) A motorbike is traveling at 85 km/hr as shown on its speedometer. How many seconds will it take to cover a 200 m track at this constant rate?

2. Average Velocity (v )

Average velocity can be determined two ways: a) by formula:

d v  v - average velocity, d – displacement, t - time t

- 2 - Physics 30S - Position – Time Graphs: Types of Velocities This formula would be used when you are asked to determine average velocity in a word problem.

e.g.1 A bike travel 3.2 km in 45 minutes. What is its average velocity?

e.g.2 A vehicle travels 30 km East in 15 minutes, 45 km West in 20 minutes then 10 km East in 7.5 minutes. What is its average velocity?

e.g. 3 A orienteering athlete walks 1.2 km East in 25 minutes, then 0.8 km North in 15 minutes. Determine her average velocity.

- 3 - Physics 30S - Position – Time Graphs: Types of Velocities b) by determining the slope of the line joining two points

In the example graph the slope of the line in time interval A would yield the average velocity for that time interval.

position (m)

4 A C ) m (

n F o i t i

s -4 o p E -8 D

-12 0 5 10 15 20 time (s)

v = slope Determine the average velocity for the rise = remaining time intervals as well as the run entire trip. 6m  0m = 2s  0s 6m = 2s = 3m/s

- 4 - Physics 30S - Position – Time Graphs: Types of Velocities

When determining the average velocity the following points should be kept in mind. a) Positive slope (rises to the right) yields positive velocity and negative slope (drops to the right) yields negative velocity. b) Positive velocity results from movement in a positive direction and negative velocity from movement in the negative direction. c) A straight-line portion on a graph indicates a constant velocity. This means that the average and constant velocities are the same value. If average velocity over more than one straight-line portion of a graph is desired, then you must construct a straight line from a point on the graph at the beginning of the time interval to a point on the graph at the end of the time interval. If d you use the average velocity formula v  , just make certain that you have t determined the total displacement over the time interval. d) A horizontal line indicates zero slope and therefore zero velocity; that is, the object is stationary.

3) Instantaneous Velocity (vinst)

There are two possible conditions when vinst is to be calculated.

a) On a straight-line graph: In this case vinst = v because velocity is constant on any straight-line graph

slope A = slope = slope C

slope C

slope B

slope A t

Any time you are asked for instantaneous velocity, and the graph is a straight line; simply find the average velocity for the corresponding interval.

b) On a curved-line graph: In this case the portion of the graph over which the vinst is to be calculated is reduced as shown in the diagram below.

slope B slope C

slope A

As a shorter and shorter time interval is chosen, shown in the diagram as slopes C, B, and A the line become progressively shorter and the curve becomes straighter between those points. If we were to choose a very short time interval so that time could be considered zero, the length of the curve would also be extremely short. Since decreasing the length of the time interval caused the portion of the graph to be straighter it would be reasonable to assume that the very short piece of graph would, for all purposes, be considered a straight line. It would be impossible to find the slope of such a short line but we can employ a construction method, which will lengthen and consequently allow us to calculate the slope of the line under consideration. The construction involves drawing a tangent line at the point where we want to find the vinst on the graph. This tangent line represents an extension of the very short piece of line that was mentioned earlier. The construction of the tangent line is shown below.

Draw the tangent line that best approximates the slope of the curved line at that point.

vinst  slope of tangent line rise  run

Practice

Find the instantaneous velocities at points A, B, and C using the graph shown below. y        B           C  A    x                 

A) Position-Time Graph For A Rolling Ball

Use the graph below to answer the following questions.

Determine:

a) Δd from 0-4 seconds f) v from 0-4 seconds

b) Δd from 4-5 seconds g) v from 6-9 seconds c) Δd from 6-9 seconds

h) vinst at 10 seconds d) Δd from 9-14 seconds

i) vinst at 5 seconds e) Δd from 0-14 seconds j) v from 0-8 seconds

position vs time

10 ) m ( 5 n o i t i

s 0 o p -5 0 5 10 15 time (s)

B) Position-Time Graph Using a Curved Line

Determine: a) v from 0-6 seconds f) vinst at 14 seconds

g) vinst at 18 seconds b) v from 8-12 seconds h) d from 0-6 seconds c) v from 12-16 seconds i) d from 16-22 seconds

d) v from 16-22 seconds j) d from 0-22 seconds

e) vinst at 7 seconds k) v from 6-16 seconds

10 ) m ( 5 n o i t i

s 0 o p -5

-10 0 2 4 6 8 10 12 14 16 18 20 22 24 time (s)

C) Position-Time Graph Reveiw

Determine:

a) the kind of motion from d) v from 6-9 seconds 0-6 seconds

e) d from 0-6 seconds b) vinst at 4 seconds

f) v from 10-11 seconds c) v from 0-6 seconds

g) vinst at 2 seconds

position vs time

n 2 o i t

o -2 p -4 0 2 4 6 8 10 12 time (s)

D) Position-Time Graph Checkup

Use the following graph to answer the following questions.

Determine:

a) Δd from 0-4 seconds

- 10 - Physics 30S - Position – Time Graphs: Types of Velocities b) Δd from 9-16 seconds d) v from 9-16 seconds

c) v from 0-4 seconds e) vinst at 3 seconds

f) vinst at 10 seconds

position vs time

n 2 o i

s -2 o

p -4 -6 0 2 4 6 8 10 12 14 16 time (s)

- 11 - Physics 30S - Position – Time Graphs: Types of Velocities ANSWERS:

Graph 1 a) +6m b) 0m c) -12m d) +4m e) -2m f) 1.5 m/s g) -4 m/s h) 0.8 m/s i) 0 m/s j) -0.25 m/s

Graph 2 a) 2 m/s b) -2 m/s c) 0 m/s d) 1.3 m/s e) -7.5 m/s f) 0 m/s g) 1.4 m/s h) 12m i) 8m j) -3m k) -2.3 m/s

Graph 3 a) variable b) 0.75 m/s c) 0.67 m/s d) -1.875 m/s e) 4m f) 3.5 m/s g) 0.4 m/s

Graph 4

a) -6m b) +6m c) -1.5 m/s d) 0 m/s e) 0.86 m/s f) -1.5 m/s g) 1.57 m/s

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