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Comeniusproject
TEWISE
"KINEMATICS"
Żyjemy w świecie ruchu
Józefina Turło
Andrzej Karbowski
Grzegorz Osiński
Krzysztof Służewski
EXPRESSTRAIN
Instytut Fizyki
Uniwersytet Mikołaja Kopernika, Toruń, Polska
COMENIUS-C2110650-CP-1-2002-AT
All rights reserved.
Privacy [email protected] for the project -team:
by Project "TEWISE" Copyright © 2002-2010
This project has been funded with support from the European Commission.
This publication [communication] reflects the views only of the author, and the Commission
cannot be held responsible for any use which may be made of the information contained
therein.
UNIFORM AND NON-UNIFORM MOTION
List of modules
I. UNIFORM MOTION
1. What we understand by uniform motion?
2. The speed of bodies
2.1. What is speed?
2.2. Measuring speed, units
2.3. Average speed
2.4. Movement formulas
3. The velocity of bodies
3.1. What is velocity?
3.2. Distance – time plot
3.3. Velocity – time plot
4. Experimental investigation of uniform motion
4.1. Measurements of distance and time for toy car
4.2. Distance – time plots
5. Questions and Tasks
II. NON-UNIFORM MOTION
1. Acceleration
1.1. What is acceleration?
1.2. Calculation of the acceleration
1.3. Examples of everyday accelerations
2. Experimental investigation of accelerated motion
2.1. Why the toy car is moving down?
2.2. School version of the historical Galileo experiment with inclined
plane
3. Road safety
4. Questions and Tasks
UNIFORM MOTION
What is uniform motion?
Could You please give some examples of uniform motion? (walking, running,
bicycling, driving, swimming, flying...)
Motion is basic phenomenon of nature. Form early infancy we sense moving objects in our
surroundings. The sensation of our own motion can provide powerful experiences. We can
recognise if the motion is uniform or accelerated.
What we understand by uniform motion?
Suppose an object moves equal distances during equal time intervals. Such motion is called
uniform motion.
Below the examples of uniform motion in everyday life are shown:
walking
windsurfing
travelling by expresstrain
sailing, flying
flying with the use of hang-glider
Aerobatics
TEWISE
„Uniform motion”
1a
The speed of bodies
What is speed?
The speed of an object is how far it goes in a unit of time. It is measured in metres per
second (m/s), or kilometres per hour (km/h). There are the units of speed. The objects are
moving with different speeds, what we can see in the table 1.
Table 1. Comparison of speed of different bodies.
Speed
light speed
Earth in orbit
typical Earth satellite
fast jet aircraft
Concorde (supersonic jet)
average speed of air molecules
sound in air
Boeing 747 Jumbo Jet
fastest bird (falcon)
high-speed train (French TGV)
motorway speed limit (UK 70 mph)
town speed limit (UK 30 mph)
Olympic sprinter
average walking speed
average speed of snail
m/s
300 000 000
29 790
7 500
833
648
500
340
270
97
60
31
13.4
10.3
1.7
0.006
km/h
1 080 000 000
107 244
27 000
3 000
2 333
1 800
1 224
970
350
216
112
48
37
6
0.02
Measuring speed, units.
Many different instruments are used to measure the wide range of speeds shown in the table 1.
But all of them need to measure just two things: time and distance travelled. You can
calculate the speed (distance travelled per second) like this:
speed in metres per second =
distance in metres
time in seconds
or using symbolic notion:
v (m/s) = s (m)
t (s)
TEWISE
„Uniform motion”
1b
The speed of car is indicated by speedometer (Fig. 1).
Fig. 1. The speedometer shows what is the speed of car.
Average speed
When you are planning a journey you can use the concept of average speed to work out how
long the journey is likely to take. We can introduce average speeds using a formula:
average speed = total distance travelled
time taken for journey
written as a formula:
v= s
t
Movement formulas
average speed =
distance travelled
time taken
distance covered = average speed x time taken
time taken =
distance travelled
average speed
TEWISE niform motion”
v= s
t
s = vt
t= s
v
1c
The velocity of bodies
What is velocity?
If you know the speed of an object and its direction of travel, then you know its velocity. The
velocity of an object is its speed in a defined direction – it is a vector quantity, thus the speed
is the value of velocity vector.
In simple situations, an object travelling in a straight line in one direction can be described as
having a positive velocity, whereas it is travelling towards it in the opposite direction will be
described as having a negative velocity. The minus sign indicates that it is travelling in the
opposite direction.
Velocity can be expressed by the formula:
distance travelled in a given direction (m)
time taken (s)
velocity (m/s) =
Distance – time plot
A graph of distance against time is called distance – time graph. The shape of the graph
enables you to see how the object is moving.
Could you draw the distance – time graph for uniform motion?
10
s [m]
9
8
7
6
5
4
3
2
1
0
0
1
2
3
4
5
6
7
8
9
10
t [s]
Fig. 2. Distance – time graph for uniform motion – pupil version
We will check your conception. Please do some experiments with uniform motion of air
bubble inside the tube of glass filled with the liquid. Measure the times, in which air bubble is
passing the marks of equal distances on the surface of glass tube.
TEWISE
„Uniform motion”
1d
Fig. 3. The glass tube with the liquid and air bubble inside
Please fill in the table 2 the distance between marks and measured times. Calculate the speed
and average speed of bubble.
Table 2.
Number .....
1
2
3
4
5
distance [m]
time [s]
speed [m/s]
average speed [m/s]
Please draw the distance – time graph for uniform motion again.
1
s [m]
0,8
0,6
0,4
0,2
0
0
0,2
0,4
0,6
0,8
1
t [s]
Fig. 4. Distance – time graph for uniform motion – second pupil version
Please note, the steeper slope of s(t) graph, the greater the speed.
TEWISE
„Uniform motion”
1e
v2
v1
α1
α2
Fig. 5. Distance – time graph for motion with constant speeds
Velocity – time plot
A plot of velocity against time is called velocity - time graph. For uniform motion velocity is
constant and the plot will be as shown below.
10
v [m/s]
8
6
4
2
0
0
2
4
6
8
10
t [s]
Fig. 6. Velocity – time plot for uniform motion
TEWISE
„Uniform motion”
1f
EXPERIMENTAL INVESTIGATION OF UNIFORM MOTION
Measurements of distance and time for toy car
Measurements of distance and time for toy car
An electrical toy car is moving uniformly along the table.
Figure 7. An electrical car is moving uniformly along the table
Please measure the time using a stop-watch and the distance travelled after each 2 seconds.
Fill in the table 3 the obtained data.
Table 3.
0
time [s]
distance [m]
2
4
6
8
10
Distance – time plots
Please make the graph of distance – time. We can learn a lot from that graph. We can tell how
far object has travelled and how fast it was moving.
TEWIS
„ Experimental investigation of uniform motion”
2a
distance [m]
3,5
3
2,5
2
1,5
1
0,5
0
0
1
2
3
4
5
6
7
8
9 10
time [s]
Fig. 8. Distance – time graph for a toy car
Please calculate the speed of the car using the below formula and write the data in the table 4:
distance moved
time taken
average speed =
Table 4.
Time [s]
speed [m/s]
2
4
6
v= s
t
8
10
What can you tell about the speed of car toy? Is it steady?
Questions and Tasks
•
•
•
•
•
•
Do you walk at a constant speed? Please measure the distance and time, calculate the
speed and make the distance-time and speed-time graphs.
Do objects fall at a constant (steady) speed?
What is the average speed of a cyclist travelling to school?
Could you calculate the average speed of flow in a river and make a speed-time graph?
How long it takes light to travel the distance of km from the Sun to the Earth, at a speed of
300 000 kilometres per second. Give the answer in minutes and seconds.
Could you calculate how distance the light travel during one year?
TEWISE
„ Experimental investigation of uniform motion”
2b
NON-UNIFORM MOTION
Acceleration
Some kinds of movement are not uniform. Can you give some examples of such motions?
(starting and stopping vehicles, kicked ball, jumping man, falling bodies in the air…). The
main feature of these motions is acceleration.
What is acceleration?
Acceleration is a rate at which the velocity of moving bodies is changing. It is measured in
meters-per-second per second, which is written as m/s2. It tells us how much the velocity is
changing during each second.
Calculation of the acceleration
Acceleration can be calculated by using the formula:
change in velocity [m/s]
time taken for the change [s]
acceleration [m/s2] =
or in symbols:
a=
v1 − v 0
t
or
a=
∆v
,
t
where vo is the velocity at the beginning of the time interval and v1 is the velocity at the end
of the time interval.
When the object is slowing down, the change in velocity is negative (because v1 is less than
vo), and so the acceleration is negative. This is sometimes called a deceleration (negative
acceleration).
Examples of everyday accelerations
The objects are moving with accelerations, what we can see in the table 1.
Table 1.
Rocket blasting into space
Object falling to the floor
Train pulling out a station
Ferry moving away from its moorings
100 m/s2
10 m/s2
1 m/s2
0.1 m/s2
In general the acceleration of an object depends on:
• the value of the force causing the acceleration,
• the mass of the object.
When a resultant force acts on an object, it speeds up (accelerate) in the direction of the force.
TEWISE
„ Non -uniform motion”
3a
Experimental investigation of accelerated motion
Why the toy car is moving down?
The toy car is moving down along the inclined table with acceleration as a result of
gravitational force. Using a stop-watch and pieces of the black buttons please make the
measurements of distance after a time intervals of 2 seconds.
Figure 1. The toy car is moving down along the table
Fill in the table 2 and table 3 the obtained data.
Table 2.
time [s]
distance [m]
0
2
4
6
Table 3.
time [s]
speed [m/s]
0
2
4
6
TEWISE
„ Non -uniform motion”
3b
distance [m]
Please make the distance-time and speed-time graphs.
1,6
1,4
1,2
1
0,8
0,6
0,4
0,2
0
0
1
2
3
4
5
6
7
8
time [s]
speed [m/s]
Fig. 2. Distance-time graph
0,3
0,25
0,2
0,15
0,1
0,05
0
0
1
2
3
4
5
6
7
8
time [s]
Fig. 3. Speed-time graph
Please calculate the acceleration using the values of speed and time taken from the table 3.
Table 4.
time [s]
acceleration [m/s]
2
4
6
Please answer – is it the motion with a steady acceleration?
TEWISE
„ Non -uniform motion”
3c
DIFFERENCE BETWEEN ACCELERATION AND DECELERATION
School version of the historical Galileo experiment with inclined plane
We can investigate motion with steady acceleration in the classroom using the idea of Galileo
historical experiment. We can use inclined table as a substitute of inclined plane, a metal ball
and a stop-watch to measure the time. Please note that Galileo used a special water clock for
measurement of time.
Figure 4. The metal ball is moving along the inclined table
Please make the necessary measurements and put the data to the below tables.
0
2
distance [m]
Table 5.
time [s]
distance [m]
4
6
8
2
1,6
1,2
0,8
0,4
0
0
1
2
3
4
5
6
7
8
9 10
time [s]
Fig. 5. Distance-time graph for a small metal ball.
TEWISE
„ Difference between acceleration and deceleration”
4a
0
2
speed [m/s]
Table 6.
time [s]
speed [m/s]
4
6
8
1
0,8
0,6
0,4
0,2
0
0
1
2
3
4
5
6
7
8
9 10
time [s]
Fig. 6. Speed-time graph for the ball.
Table 7.
2
4
6
8
time [s]
acceleration [m/s]
You can plot the distance-time, speed-time and acceleration-time graphs and explain what
kind of motion was investigated in this experiment.
TEWISE
„ Difference between acceleration and deceleration”
4b
Road safety
It takes an alert driver about 0.7 s to apply the brakes in response to seeing a hazard. This time
is know as the reaction time. The time does not depend on the car’s speed. A car travelling at
100 km/h will travel twice as far in 0.7 s as one travelling at 50 km/h so the reaction distance
is twice as much.
The breaking distance is the distance the car will travel between the time the brakes are
applied, and the time that if finally stops. Doubling the speed increases 4 times the braking
distance.
The greater the speed of the car:
• the greater the breaking force needed to stop it in a certain time,
• the longer the time needed to stop it with a certain braking force, and so the further it
travels before it stops.
When roads are wet or icy the braking distance is greater.
The increase of stopping distance as the response to increase of speed is shown in the Fig. 7.
32
km/h
48
km/h
64
km/h
80
km/h
96
km/h
112
Km/h
Fig. 7. The stopping distance is the reaction distance plus the braking distance*
* G. Dolan, M. Duffy, A. Percival, Physics, Heinemann Coordinated Science, UK, 1996.
TEWISE
„ Difference between acceleration and deceleration”
4c
Questions and Tasks:
•
•
•
•
•
Please give some examples of motion with steady acceleration from the everyday life?
A train accelerates at a steady rate from rest for 30 seconds, by which time has reached a
velocity 15 m/s. Please draw the speed-time graph and use the data to calculate:
a) the train acceleration,
b) the distance traveled in the first 30 seconds of its journey.
How long would a car take to increase its speed from 8 m/s to 24 m/s at a constant
acceleration of 3,2 m/s2?
Could you explain what kind of motion do you observe when a bus is starting and next
moving and stopping at the bus stop.
When a car stops suddenly, the stopping distance is the sum of the reaction distance and
the braking distance. Which of these two factors might be affected by a slippery road
surface? Explain why both factors might be affected if the driver had consumed alcohol or
drugs.
TEWISE
„ Difference between acceleration and deceleration”
4d
Project Tewise
Forces – why things are moving?
When bodies are at rest?
Forces
The most simple forces are pushes and pulls. If we push or pull on an object, it often moves.
Sometimes the force makes the shape of an object change.
A force can start an object are moving. It can also slow down or speed up a moving object.
Sometimes a force seems to be doing nothing. This might be because it is cancelling out the
effect of another force.
How forces can be measured?
To measure forces we use a newton meter. This is a spring and a scale. The scale is calibrated
in newtons. The newton N is the unit of force. It is named after Sir Isaac Newton (1642 1727).
Picture 1. The newton meter can measure the force.
When talking about forces we should therefore specify not only their size, but also their
direction. The direction might be described as "to the left" or "to the right", or as "acting in an
opposite direction". Quantities such as forces which have a direction associated with them are
called vector quantities.
TEWISE
„Forces – why things are moving?”
5a
Project Tewise
Forces – Newton’s First Law
What are the reasons that objects are moving?
How we can stop the moving bodies?
Weight is a force
The weight of an object is simply the force that acts on it because of the planet’s gravity. It
always acts in a downwards direction (towards the centre of the planet). Weight and mass are
related by the formula:
weight (N) = mass (kg) * Earth acceleration (m/s2)
or in symbols:
W = m*g ,
where g = 9.81 m/s2 .
Forces change the way things move
Sir Isaac Newton spent a lot of time thinking about forces. He stated some important laws
about them. Newton’s First Law says that an object keeps on going as it is, unless an
unbalanced force acts on it.
This helps describe what a force is. A force doing something that changes the way in which
an object moves. The force will either speed it up or slow it down.
Experiment
Put a book on the table. If you give it a push, it starts moving. Newton’s First Law says that
when you stop pushing, the book will keep on going – unless it is acted on by an unbalanced
force. When you stop pushing, the book very quickly stops moving.
Examples
If all the forces acting on a moving object are balanced it will keep on moving with steady
speed.
Picture 2. The air resistance force and weight acting of a parachutist can be balanced.
TEWISE
„Forces – why things are moving?”
5b
Project Tewise
Forces – Newton’s Third Law
One force always produces a reaction to itself
If a book is placed on a table, the force of its weight acts downwards. So why does it not
move downwards? A simple answer would be that the table gets in the way.
Picture 3. The action and reaction forces.
Newton’s Third Law states that if one body pushes on a second body, the second body
pushes back on the first with the same forces. For every action there is an equal opposite
reaction.
If the book pushes down on the table, the table pushes up on the book, so the book does not
move. The table rests on the Earth and the Earth pushes back with an equal and opposite
force. So the table stays where it is.
Experiment
Please connect two newton meters and pull them outside. Check the values of action and
reaction forces. Are they equal and opposite?
Picture 4. Action and reaction forces act between two newton meters.
TEWISE
„Forces – why things are moving?”
5c
Project Tewise
Forces – Newton’s Third Law
Experiment
Please use two magnets to check if the forces are created in pairs?
Picture 5. Magnets attract and repel one another with equal and opposite forces.
The effects of the forces may be more apparent in the case of the tennis racquet and ball. The
force of the racquet on the ball stops its motion and propels it away again. The force of the
ball on the racquet deforms the racquet strings.
Picture 6. As the racquet hits the ball, an equal and opposite force affects the racquet.
Questions and Tasks:
•
•
•
•
•
What forces are acting on a book resting on a table?
Are the forces of action and reaction in equilibrium cancelling themselves?
What is the weight of a bag of shopping of total mass 5 kg?
What is the mass of a pupil of weight 500 N?
What would happen to your mass and your weight if the Earth’s gravitational field
strength were to double?
TEWISE
„Forces – why things are moving?”
5d
Project Tewise
Experiments, tables and plots concerning falling bodies
What are the reasons that objects are moving with an acceleration?
When bodies can moving with the steady acceleration?
Force and acceleration
The relationship between acceleration, force and mass is given by the formula:
acceleration (m/s2) =
force (N)
mass (kg)
or in symbols:
a=
F
.
m
Acceleration increases with the force and decreases with the mass.
When no resultant force is acting, there is no acceleration, and therefore no change in
velocity. A resultant force produces an acceleration proportional to the force. This formula
can be rearranged to find the force needed to give an object a particular acceleration:
F = ma .
From the formula above it can be seen that one newton (1 N) is the force needed to give a
mass of 1kg an acceleration of 1 m/s2:
1 N = 1 kg * 1 m/s2 .
Weight and acceleration
Experiment
Can object freely fall down in an evacuated tube (one where the air has been removed) with
the steady acceleration? What is the reason of this?
Expresstrain 2004
„Experiments, tables and plots concerning falling bodies”
6a
Project Tewise
Forces – Newton’s Third Law
Experiment
Please use two magnets to check if the forces are created in pairs?
Picture 5. Magnets attract and repel one another with equal and opposite forces.
The effects of the forces may be more apparent in the case of the tennis racquet and ball. The
force of the racquet on the ball stops its motion and propels it away again. The force of the
ball on the racquet deforms the racquet strings.
Picture 6. As the racquet hits the ball, an equal and opposite force affects the racquet.
Questions and Tasks:
•
•
•
•
•
What forces are acting on a book resting on a table?
Are the forces of action and reaction in equilibrium cancelling themselves?
What is the weight of a bag of shopping of total mass 5 kg?
What is the mass of a pupil of weight 500 N?
What would happen to your mass and your weight if the Earth’s gravitational field
strength were to double?
TEWISE
„Forces – why things are moving?”
5d
Project Tewise
Experiments, tables and plots concerning falling bodies
Picture 2. A book and page of paper fall downwards.
To check the kind of motion of a book we propose to use Data Video module from Coach 5
software. Data Video allows to create activities to do interactive video measurements. We can
collect position and time data from digital video clips in the form of points (see Picture 3).
Picture 3. Position of a book marked in the video clips.
This data can be plotted in a graph, viewed in a table and the software can calculate the
velocity and acceleration of a moving object.
TEWISE
„Forces – why things are moving?”
6c
Project Tewise
Experiments, tables and plots concerning falling bodies
Picture 3. Position of the book in function of time.
Picture 4. Velocity of the book in function of time.
Please read the velocity and time values from the above graph (see Picture 4), calculate the
acceleration and prepare the acceleration-time graph. Can you explain what kind of motion is
this? Have this object steady acceleration?
7. Questions and Tasks:
•
If the mass of astronaut is 65 kg, calculate her weight on the Earth and what it would be
on the Moon.
• Calculate the acceleration of a freely falling coin and a feather on the Moon.
• If the thrust produced by a rocked is constant, how will its acceleration change with time?
Explain your answer.
• A car hits a brick wall at 15 m/s. The mass of the car is 1000 kg. The car is stopped in
0.5 s:
a) what is the deceleration (negative acceleration) of the car?
b) what average force does the wall exert on the car?
c) what average force does the car exert on the wall?
Expresstrain 2004
„Experiments, tables and plots concerning falling bodies”
6d
Project Tewise
Forces and linear motion in real life
Investigation of bicycle's motion
In this activity we are going to investigate a motion of a bicycle during speeding up and
slowing down. Using Data Video (from Coach 5 software) we would like to know how
velocity and acceleration changes and what kind of motion we can recognize on the below
graphs. First we should measure the position of the cyclist choose as a video point location
(see Picture 1).
Picture 1. Trace of the bicycle's motion.
What can you tell about the kind of cyclist motion? To answer this question we propose to
analyze position time graph and velocity time graph.
Picture 2. Position of cyclist in funtion of time.
TEWISE
„Forces and linear motion in real life”
7a
Project Tewise
Forces and linear motion in real life
Picture 3. Horizontal velocity of cyclist in funtion of time.
What was the initial velocity of the cyclist?
What was the final velocity of the cyclist?
What can you deduce about the cyclist's acceleration?
Can you calculate the acceleration in the middle of the distance?
Calculate the average acceleration of the cyclist by calculating the change in velocity divided
by the corresponding change in time.
Bicycle slowing down
In that case we would like to investigate the bicycle's motion study the below graphs.
Picture 4. Position of cyclist marked in video clip.
TEWISE
„Forces and linear motion in real life”
7b
Project Tewise
Forces and linear motion in real life
Picture 5. Position in function of time.
Picture 6. Horizontal velocity in function of time.
Picture 7. Acceleration in function of time.
What can you tell about this kind of motion? Do you know why the bicycle slowing
down?
TEWISE
„Forces and linear motion in real life”
7c
Project Tewise
Forces and linear motion in real life
The motion of the basketball
Basketball shot
The motion of the basketball is an example of projectile motion. The motion can be divided
into two parts: the horizontal component and vertical component. These two components can
be calculated independently for each other and then the results can be recombined to describe
the total motion.
Picture 8. Position of basketball marked in video clip.
After the basketball shot the only forces acting on the ball are the forces of gravity and air
resistance. The force of air resistance depends on the ball's velocity.
Picture 8. The horizontal and vertical positions of the ball.
Can you explain what kind of motion we have in horizontal and vertical directions?
Please calculate the average acceleration of a basketball.
TEWISE
„Forces and linear motion in real life”
7d
Project Tewise
Summary
Displacement
This is the distance moved in a defined direction. Displacement is a vector quantity.
Speed
The speed of an object is how far it goes in a unit of time.
Velocity
The rate of change of displacement. Velocity is a vector quantity, It is not the same as speed
as the latter does not take acoount of direction an so it not a vector quantity.
Acceleration
The rate of change of velocity. Acceleration is most commonly thought of in terms of an
increase in speed and deceleration a decrease in speed, thought this ignores the fact, that
direction is also involved.
Uniform motion
Motion in which an object moves equal distances during equal time intervals.
Newton's first law of motion
Every object remains at rest or in uniform motion in a straight line unless acted upon by an
unbalanced force.
Newton's second law of motion
The relationship between an object's mass m, its acceleration a, and the applied force F is:
F = ma.
Acceleration and force are vectors. In this law the direction of the force vector is the same as
the direction of the acceleration vector.
This is the most powerful of Newton's three Laws, because it allows quantitative calculations
of dynamics: how do velocities change when forces are applied.
Newton's third law of motion
If a body A exerts a force on body B, then body B exerts an equal an opposite force on body
A.
The law expresses a very simple idea: that all forces occur in pairs, equal in size, but acting in
opposite directions and on different objects.
TEWISE
„Summary”
8