III.1. Movement and rest.
- III.1.1. Reference system. Movement and rest.
- III.1.2. Traveled distance. Movement duration.
- III.1.3. The average speed.
- III.1.4. Types of movement.
- III.1.5. The graphical representation of the movement.
- III.1.6. Average acceleration.
- III.1.7. The uniformly varied rectilinear motion.
III.1.1. Reference system. Movement and rest.
A body with dimensions negligible to the size of the displacements or to the distance at which other bodies are located can be represented by a point called the material point.
The mobile is a body that can move and is represented by a material point.
The landmark (reference body) is a body against which the position of another body is determined. It serves to spatially locate a body.
Time measuring instruments (clock, stopwatch) are used for temporal location.
A body is in a state of motion when the body changes its position (or distance) from the chosen landmark.
A body is at rest when the body does not change its position (or distance) from the chosen landmark.
E.g:
You're in the physics lab, in the bank, in physics class. What mechanical condition are you in, moving or at rest?
If you choose as a landmark: the bench, the wall, the blackboard, etc. you are at rest, because you do not change your distance from these landmarks.
If you choose as a landmark: a traveler on the street, a bird flying out the window, the Sun, the Moon, etc. you are in a state of motion, because you change your distance from these landmarks.
We say that movement and rest are relative states, because they depend on the chosen landmark. So we can be in a certain moment of time both in motion and at rest depending on the chosen landmark.
But we cannot say about a body that it is both moving and at rest, at a certain moment and compared to the same landmark.
There is no absolutely fixed body (landmark) in the whole Universe, because there is no body at absolute rest. All bodies are moving relative to each other, because in reality, the chosen landmark is itself moving relative to other landmarks.
For example, I am resting in front of her in the car. But the car is moving against the road. The road moves towards the Earth with it. The earth moves toward its own axis and toward the Sun and other celestial bodies. Our solar system is moving toward our Galaxy, the Milky Way. Our galaxy is moving relative to the other galaxies in the universe.
But when we study the movement of a mobile with respect to a reference system (denoted SR), we consider its fixed landmark.
The trajectory is the line described by a mobile moving against a landmark (the line obtained by joining all the points that constitute the successive positions of the mobile).
Trajectory classification:
1. The rectilinear trajectory is a straight line.
Example:
- The trajectory of the car towards the ground;
- The movement of the plunger of a syringe relative to the syringe;
- The movement of a runner in the 100 meters race etc.
2. The curvilinear trajectory is a curved line.
Example:
- The trajectory of a ball thrown to the ground;
- The movement of a red lamp fixed on the wheel of the bicycle, etc.
3. The circular trajectory is a circle.
Example:
- The trajectory of a wheel relative to its axis (its center);
- The trajectory of a cabin in the amusement park in relation to its axis;
- The movement of a red lamp fixed on the wheel of the bicycle with respect to the axle, etc.
4. The point trajectory is a point: .
Example:
- The trajectory of any body towards itself.
🔓 Solved problems
1. You are on the seat of the bus that takes you to school. In what mechanical state (movement or rest) are you compared to:
a) The bus driver? I am at rest with the driver, because I do not change my position (distance) from him.
b) Bench? I'm at rest on the bench, because I'm not changing my position on it.
c) Earth? I am moving towards the Earth, because I am changing my distance from it.
d) Cars on the road? I am moving towards cars because I am changing my position towards them.
2. Draw the trajectories of a point on the wheel of your bicycle with respect to:
a) wheel axle (center).
The trajectory is circular (circle):
b) soil (Earth).
The trajectory is curvilinear:
III.1.2. Traveled distance. Movement duration.
To calculate the distance traveled by a mobile, you need to set a reference point, called the origin.
- If the origin is right in the landmark, we denote by O (that is, x0 = 0 m).
- If the origin is not exactly in the landmark, we denote by another point, A (for example, x1 = 20 km from the landmark).
The position of a mobile on a trajectory is the distance from the origin to the mobile, measured on the trajectory. It is denoted by x.
The distance (denoted by d or Δx) traveled by a mobile is the length of the road traveled by the body from a landmark.
d = Δx = x2 – x1, dacă x2 > x1 (the mobile moves away from the landmark)
d = Δx = x1 – x2, dacă x1 > x2 (the mobile is approaching the landmark)
🔦 Remark
The distance is always positive, so we subtract the lower position from the higher position.
The duration of the movement (denoted by Δt or t) represents the time interval in which the mobile traveled a certain distance.
Δt = t2 – t1
t1 = the moment of the beginning of the movement and
t2 = the moment when the movement ends.
🔓 Solved problem
1. A cyclist starts at kilometer 20 at 12:00 and arrives at kilometer 60 at 13:30. How far did the cyclist travel and how long did the movement last?
We write the data of the problem:
Initial position: x1 = 20 km
Final position: x2 = 60 km
Initial time: t1 = 12:00
Final time: t2 = 13:30
We apply the distance and duration formula and replace the problem data:
d = Δx = x2 – x1 = 60 km – 20 km = 40 km
Δt = t2 – t1 = 13:30 – 12:00 = 1h 30min = 1,5 h
III.1.3. The average speed.
The average speed (vm) is the physical parameter equal to the ratio between the distance traveled (d) and the duration of the movement (Δt).
Characterization of the average speed as physical parameter:
🔦 Remark
a) The instantaneous speed is the speed of the mobile at a certain moment indicated by the speedometer.
For example, a car is traveling at an average speed of 70 km / h. This does not mean that the whole car had this speed. The speedometer may even indicate 140 km / h or 20 km / h.
b) The average speed is not the arithmetic average of the momentary velocities of the mobile.
c) Speed has in addition to numerical value the orientation, ie direction and sense.
The direction of the speed is given by the right on which the mobile is moving.
The sense of the speed is represented by an arrow.
The girl in the picture is moving at a speed of 1,2 m/s, in a horizontal direction, heading to the right. Indicating the numerical value (1,2), the unit of measurement (m/s), the direction (horizontal) and the sense (to the right), we completely characterized the girl speed.
📝 Minimum and maximum speed records
The snail has a speed of 1,5 cm/min (0,0009 km/h).
The speed of a Jamaican athlete on the 100 m track is 9,58 m/s (34,5 km/h).
Fanned fish can reach a speed of 110 km/h.
The cheetah can reach a speed of 113 km/h.
The peregrine falcon holds the record in the world of birds with a speed of 320 km/h.
A Japanese train with magnetic levitation reached a record speed of 603 km/h.
X-15, a fighter jet holds the record for the fastest man-made aircraft, reaching a speed of 6.7 Mach (8200 km/h).
The absolute speed record is held by light, with the highest speed of 300.000 km/s (1.080.000.000 km/h).
🔓 Solved problems
1. A mobile starts at kilometer 100 at 10:15 and arrives at kilometer 275 at 11:50. At what average speed did the mobile move in m/s?
We write the data of the problem:
x1 = 100 km
t1 = 10:15
x2 = 275 km
t2 = 11:50
vm = ? m/s
Solution:
We write the average speed formula, we calculate the distance traveled and the duration of the movement in SI and we replace in the formula:
2. A driver leaves kilometer 240 at 10:50 and arrives at kilometer 80 at 11:40. What was the average speed of the driver in m/s and km/h?
We write the data of the problem:
x1 = 240 km
t1 = 10:50
x2 = 80 km
t2 = 11:40
vm = ? m/s, km/h
Solution:
We write the average speed formula, we calculate the distance traveled and the duration of the movement in SI and we replace in the formula:
We transform the speed from m/s to km/h:
3. Transforms the following speeds into m/s:
Solution:
4. Diana lives at a distance D = 2 km from the school. One day she left home at 7:10. After covering a quarter of a distance, he realizes that he has forgotten his physics project and returns to pick it up. He arrives at school at 7:50. Determine Diana's average speed in m/s and km/h.
We write the data of the problem:
D = 2 km = 2000 m
t1 = 7:10
t2 = 7:50
If you can remember, a speed expressed in m/s can be transformed into km/h, by multiplying its value by 3,6.
🔐 Homework
1. Miruna leaves Sinaia, a city located 120 km from kilometer zero in Bucharest, at 12:55 and arrives 10 km from Bucharest at 13:50. What was his average speed?
2. Transforms the following speeds into m/s?
3. How to measure your running speed on a 100 m track:
You can ask your sports teacher or a colleague to time you, when you run two laps of the gym (after a pre-warm-up, so as not to risk any muscle strain).
Divide the distance (96 m) by the time obtained (transformed into seconds) and this is how you get your average running speed.
You can compare it to the speed record (9,58 m/s).
4. Determine the average speed at which you move, according to the below model.
III.1.4. Types of movement.
Types of movement
A) Classification of the movement according to the trajectory:
Rectilinear motion, in which the trajectory is a straight line.
Curvilinear motion, in which the trajectory is a curved line.
Rotational motion, in which the trajectory is a circle or a circular arc.
B) Classification of movement by speed:
Uniform motion, in which the speed is constant (its value does not change).
Varied accelerated movement, in which the speed increases.
Various braking movement, in which the speed decreases.
Uniform rectilinear motion is the motion in which the trajectory is a straight line and the speed is constant.
In uniform rectilinear motion:
The mobile travels equal distances in equal time intervals.
The current speed is equal to the average speed.
Speed maintains its numerical value, direction and sense of movement.
To determine the law of uniform rectilinear motion, we start from the velocity formula, noting with x0 = initial position, with x = final position, with t0 = initial moment and with t = final moment.
The law of uniform rectilinear motion:
x = x0 + v (t - t0) when the mobile moves away from the landmark
x = x0 - v (t - t0) when the mobile is approaching the landmark.
🔓 Solved Problem
1. Luiza's coach timed her movement on the 100 m track and entered the movement data in the following table.
a) What kind of movement does Luiza have?
b) Calculate the average speed.
c) Determine the law of motion.
Solution:
a) We notice in the table that Luiza travels equal distances (25 m) in equal time intervals (3 s). So Luiza has a rectilinear and uniform movement. Attention, if the body has a constant speed, the trajectory must be rectilinear. In curvilinear and rotational motion, the speed cannot be constant.
b)
When speed gives us with period, it is good to work with an irreducible fraction
c) We replace in the law of motion x0, v and t0 :
🔐 Homework
1. In the following table the data of the movement of a mobile are entered.
Is required:
a) What kind of movement does the mobile have? Why?
b) Calculate the average speed.
c) Determine the law of motion.
III.1.5. The graphical representation of the movement.
👀 The graphical representation of a mobile movement.
In order to represent the graph of the movement of a mobile (Luiza's running on the 100 m track) the following four steps must be completed:
1) Make a table with the movement data (their corresponding positions and times).
2) Draw the axes, two perpendicular lines, one horizontal and one vertical, on a math sheet or on graph paper. The horizontal axis is called the abscissa axis (Ox) and the vertical axis is called the ordinate axis (Oy). The two axes are graded (calibrated) by choosing a standard for each, so that we can represent all the positions of the mobile and their corresponding moments.
For our example I will choose as a standard (scale):
For the distance axis:
1 cm : 5 m
For the time axis:
1 cm : 1 s.
The point of intersection of the axes is O = the origin for each axis, ie at this point we have 0 m and 0 s.
The abscissa is the axis of time. Note the axis and at its end, write t (s).
The order is the distance axis (the positions of the mobile are noted on this axis). Note the axis and at its end write x (m).
3) Draw each point in the table on the graph.
The first point is right at the origin, O (x0=0, t0=0).
To represent the second point, A (x1 = 25 m, t1 = 3 s), we proceed as follows: next to the numerical value of 3 s, a vertical line goes, dotted. Next to the numerical value of 25 m, a horizontal line is pointed. The point of intersection of the two dotted lines is the point on the coordinate graph A(x1, t1).
Continue with the other three points: B (x2, t2), C (x3, t3), D(x4, t4), according to the model shown above, until all five are plotted points in the table.
4) Draw the graph of the movement by joining all the previously built points.
Remark:
Don't confuse the trajectory of the mobile with the graph of the movement!
Using the motion graph we can find more information about how a mobile moves:
- The graph of rectilinear and uniform motion is a straight line.
- If the mobile is at rest, the motion graph is a horizontal line.
- To draw the graph of the motion it is enough to represent at least two points that together, will determine the right of the graph.
- The straight line representing the rectilinear and uniform motion is more inclined (forming a larger angle with the horizontal) when the speed is higher.
🔓 Solved problems
2. The graph of the movement of a mobile is represented in the following figure:
a) What are the positions of the mobile at the moments: t1 = 0 h, respectively t2 = 2 h?
b) Is the mobile moving away or closer to the landmark?
c) When does the mobile reach the 120 km position?
d) When does the mobile reach the landmark?
e) What is the speed of the mobile in m / s?
Solution:
a) At t1 = 0 h, the mobile is in position x1 = 200 km. From origin, we climb vertically (position axis) until we intersect the graph and thus determine the position x1 = 200 km. At t2 = 2 h, the mobile is in position x2 = 100 km. We climb vertically (position axis) until we intersect the graph and from the graph we go horizontally dotted, until we intersect the position axis, so we determine the position x2 = 100 km.
b) The mobile is approaching the landmark (position 0 km), because the positions of the mobile decrease with increasing time.
c) At 120 km, the mobile is 1.5 hours after departure. We start with a dotted line of 120 km horizontally, until we meet the graph and from here, we descend dotted vertically on the time axis.
d) In the landmark, position 0 km, we have a time of 4 h. Originally, we go horizontally (time axis) until we intersect the graph and thus determine the time of 4 h.
e) We write the speed formula:
When calculating the distance, we subtract x1 - x2, because x1 > x2 (the distance is always positive).
When calculating time, we subtract t2 - t1, because t2 > t1 (time is positive).
3. In the following table the data of the movement of a mobile are entered.
Represents the graph of motion using as standards
1 cm : 2 s
1 cm : 2 m.
Solution:
4. The following table gives the data of the movement of two mobiles M1 și M2.
a) Determine the positions of the mobile M2 knowing the law of motion xM2 = 30 + 12∙ t
b) Make the graphs of the motion of the two mobiles.
Standards:
1 cm : 30 m
1 cm : 5s
c) What is the position of the two mobiles at t = 20 s?
d) Determine the average speed of the two mobiles.
e) Write the law of motion for the mobile M1.
Solution:
a) xM2 = 30 + 12 ∙ t
x0M2 = 30 + 12 ∙ 0 = 30 m
x1M2 = 30 + 12 ∙ 5 = 90 m
x2M2 = 30 + 12 ∙ 30 = 390 m
x3M2 = 30 + 12 ∙ 40 = 510 m
b) The graph of movement of the two mobiles:
c) For M1: la t = 20 s we have the position x = 120 m.
For M2: la t = 20 s we have the position x = 270 m.
d)
e) xM1 = x0 + v(t – t0) = 0 + 6 ∙ t = 6t
🔐 Exercises
1. In the following table the data of the movement of a mobile are entered.
Is required:
a) Knowing the law of motion of this mobile: x = 10 + 5 ∙ t, determine the positions of the mobile corresponding to the times in the table.
b) Represent the graph of motion using as standards
- 1 cm : 10 m
- 1 cm : 4 s.
c) Determine on the graph the position of the mobile at t = 8s.
d) Calculate the average speed.
e) What kind of movement does the mobile have? Why?
III.1.6. Average acceleration.
Uniform rectilinear movement is possible in everyday life, only on small sections of the highway. In the case of real movements, the speed of the furniture increases or decreases over time, due to traffic lights, curves, pedestrian crossings, speed limits. All cars have an accelerator pedal and a brake pedal.
A mobile has an accelerated movement when its speed increases over time.
Here are the positions of a mobile at equal intervals in accelerated motion:
At start-up, the mobile has an accelerated movement, because its speed increases from zero.
Here are the positions of a mobile at equal intervals in the braked motion:
When stopped, the mobile has a slow motion, as its speed drops to zero.
The average acceleration (am sau a) is the physical parameter equal to the ratio between the variation of the speed (its increase or decrease) and the duration in which this variation occurred.
Characterization of average acceleration as physical parameter:
III.1.7. The uniformly varied rectilinear motion.
The uniformly varied rectilinear motion is the motion of a mobile on a rectilinear trajectory with constant acceleration.
In uniformly accelerated rectilinear motion:
The mobile travels longer and longer distances at equal intervals.
The momentary acceleration is equal to the average acceleration, being constant.
Acceleration and speed have the same direction and sense as that of motion.
🔦 Remark:
The best known example of uniformly accelerated rectilinear motion is the free fall of bodies to Earth. All bodies fall to Earth with a constant acceleration, called gravitational acceleration, g = 9,81 m/s2. When the body falls free, its initial velocity is v0 = 0 m/s.
🔓 Solved problems
1. Look carefully at the data about the girl's movement. How do we determine the type of movement?
We notice that in equal time intervals (4 s), the girl travels longer and longer distances. If we calculate the speeds on each portion, they increase. The speeds increase with equal values (Δv = 1 m / s) in equal time intervals (Δt = 4 s). As the acceleration is the ratio between the variation of the speed over time, it follows that the motion is uniformly accelerated rectilinear.
2. In order to overtake, a car traveling at a speed of 54 km / h accelerates by 4 m/s2. How fast is the car after 5 s ?
Solution:
- We write down the data of the problem and turn them into SI:
- We write the formula for acceleration and remove the unknown, v2:
🔐 Homework
1. A body is allowed to fall freely from a certain height with an acceleration equal to 10 m/s2. After how long does the body reach a speed of 4 m/s?