Motion in one Dimension - Constant Acceleration
Motion in one Dimension - acceleration
In this time, I will introduce new physical quantity to you. Main character is "acceleration". It has also direction. So it's a vector physical quantity.
Figure 2-1
let's check the car's motion again.
We've defined above figure by one dimension motion.
What is the main object in the motion? What is moving now?
Yes. It's the car including the driver.
What kind of physical quantity does it have?
(1) It has 200 kg mass including him. Mass is a scalar.
(2) It has velocity as a vector. The magnitude is 20 m/s, and the direction is ( + ), east or right direction.
(3) It has displacement as a vector. The magnitude is 10 ( km ) from his house based. The direction is ( - ), west or left direction.
We've discussed about above 3 physical quantities.
(4) And especially I assumed that there is no resistances, no refractions.
At this case with (1),(2),(3),(4) conditions, now the driver stepped on the gas pedal (accelerator) softly and constantly.
Then what will happen with the car?
Surely it will get more speed. (Specifically speaking speed means the magnitude of velocity.) and so more faster gradually.
It's a significant change in motion. Stepping on the gas pedal is a normal action in our life. But in physics it's an amazing event. The motion(movement) of (a) matter(s) or (a) particle(s) or (an) object(s) is one of the main interests in physics. Then what do you think what is the representative physical quantity for motion of a matter? Maybe all of you could guess it. Yes, it's velocity.
Therefore the change of velocity is a very exciting event in physics.
However something is needed to change velocity.
That is force. In Figure 2-1, the car's velocity have been changed by stepping accelerator. We can tell that by stepping accelerator the driver gave his car a force physically.
Force is very important in motion because it makes velocity change. But I will not talk about it in detail now.
So we can conclude that where there is change of velocity there is force, and vice versa.
Velocity is a vector physical quantity, isn't it? And vector has magnitude and direction. So there are two kinds of changes for velocity. One is the change of magnitude and another is the change of direction.
Here, in our theme, the car's velocity has only the change of magnitude.
Now Let's check about how is it changing. The magnitude of velocity ,in other words, speed is changing with respect to time.
I want to emphasize the phrase, "change with respect to time".
There is a concept called "rate of change". And so there are many rates of change with respect to physical quantities. Typically rate of change with respect to time is best popular. Another thing, for instance, there is rate of change with respect to space. But most of "rate of change" is about "with respect to time".
The rates of change generate driven physical quantities. And driven physical quantities have special physical units.
Let me introduce one more thing for physics.
To describe nature by physics we usually use 3 tools frequently.
(1) Mathematics
(2) Graph
(3) Unit
Unfortunately mathematics is very important although it's hard to study. So many physical laws are expressed by mathematical formulas. We can not escape from math to study physics.
And Drawing graph is a also good tool for physics. Let's see below graph.
Figure 2-2
We can see the status of speed according to time change. We assumed the driver stepped an accelerator at 5:30 pm. So from that time the speed will increase continuously. We can draw a graph like above, and we can understand the car's motion with the graph.
3rd tool, unit, the physical unit is also very important to describe physical phenomena. As I told you when we see a physical unit we can guess the meaning. Look into the unit of velocity. It's (m/s). It means one meter distance moved per second. So we can understand that the definition of velocity. It's the rate of distance with respect to time. Exactly speaking it's the rate of displacement with respect to time. Distance moved is the same with the magnitude of displacement.
I will describe it by mathematics.
velocity
It looks difficult. But don't worry. I'll skip explanation for now.
This time, let's focus on the acceleration. Today's theme is the car's acceleration. However please keep in mind the other explanations.
When we see the car's motion, it becomes faster continuously. Because the accelerator has been stepped softly continuously. The driver doesn't loose it. It's called a constant acceleration motion. The acceleration will not be changed. Its magnitude and direction will not be changed.
We can guess acceleration is related to velocity. Let's check its unit.
m/s2
acceleration
So, if acceleration is 1m/s2, and initial speed is 20 m/s, then the speed will be 21 (m/s) after 1 second. After 10 second the speed will be 30 m/s.
displacement or distance, velocity or speed, constant acceleration, time, between those 4 physical quantities, there are physical formulas
(1) V = V0 + at
(2) S = V0 t + 1/2 * at2
(3) 2aS = V2 - V02
( t: time to be taken for S distance moving of the car ,
a: constant acceleration ,
S: distance moved after time t ,
V0: initial velocity ,
V: velocity after time t, or velocity when the car moves S distance )
Actually we have to study them deeply with using mathematics. We have to learn the concept of differentiation and integral calculus. But I will skip them. I hope later we can look into them later.
Anyway by stepping on the gas pedal (accelerator), the car got constant acceleration toward (+) direction. It's the same direction with velocity. Therefore the speed will be bigger and bigger continuously.
So we can guess the driver may arrive more earlier for home. Then let's calculate the arrival time.
The car is on (-) 10 (km) position when we set his house by origin (0 point). In other words he have to go 10 km more for his home. And the constant acceleration is 1 m/s2 , the initial velocity is 20 m/s, both of them are (+) direction.
To calculate the arrival time, I will use 3rd formula.
Be careful, to calculate we need to convert unit by SI(mks) unit.
10 km is equal to 10 x 10^3 (m), that is 10 x 1000 (m).
2aS = V2 - V02
2x1x10x10^3 = V2 - (20)2
V= 140 (m/s)
We've calculated the speed of the car when it has arrived at home.
Again with using 1st formula,
V = V0 + at
140 = 20 + 1 * t
t = 120 (s) = 2 x 60 (s) = 2 (minute)
So he will get to home at 5:32 (pm).
HW 2-1) Drive the 3rd formula for constant acceleration motion with using formula 1 and 2.
In this time, I will introduce new physical quantity to you. Main character is "acceleration". It has also direction. So it's a vector physical quantity.
Figure 2-1
let's check the car's motion again.
We've defined above figure by one dimension motion.
What is the main object in the motion? What is moving now?
Yes. It's the car including the driver.
What kind of physical quantity does it have?
(1) It has 200 kg mass including him. Mass is a scalar.
(2) It has velocity as a vector. The magnitude is 20 m/s, and the direction is ( + ), east or right direction.
(3) It has displacement as a vector. The magnitude is 10 ( km ) from his house based. The direction is ( - ), west or left direction.
We've discussed about above 3 physical quantities.
(4) And especially I assumed that there is no resistances, no refractions.
At this case with (1),(2),(3),(4) conditions, now the driver stepped on the gas pedal (accelerator) softly and constantly.
Then what will happen with the car?
Surely it will get more speed. (Specifically speaking speed means the magnitude of velocity.) and so more faster gradually.
It's a significant change in motion. Stepping on the gas pedal is a normal action in our life. But in physics it's an amazing event. The motion(movement) of (a) matter(s) or (a) particle(s) or (an) object(s) is one of the main interests in physics. Then what do you think what is the representative physical quantity for motion of a matter? Maybe all of you could guess it. Yes, it's velocity.
Therefore the change of velocity is a very exciting event in physics.
However something is needed to change velocity.
That is force. In Figure 2-1, the car's velocity have been changed by stepping accelerator. We can tell that by stepping accelerator the driver gave his car a force physically.
Force is very important in motion because it makes velocity change. But I will not talk about it in detail now.
So we can conclude that where there is change of velocity there is force, and vice versa.
Velocity is a vector physical quantity, isn't it? And vector has magnitude and direction. So there are two kinds of changes for velocity. One is the change of magnitude and another is the change of direction.
Here, in our theme, the car's velocity has only the change of magnitude.
Now Let's check about how is it changing. The magnitude of velocity ,in other words, speed is changing with respect to time.
I want to emphasize the phrase, "change with respect to time".
There is a concept called "rate of change". And so there are many rates of change with respect to physical quantities. Typically rate of change with respect to time is best popular. Another thing, for instance, there is rate of change with respect to space. But most of "rate of change" is about "with respect to time".
The rates of change generate driven physical quantities. And driven physical quantities have special physical units.
Let me introduce one more thing for physics.
To describe nature by physics we usually use 3 tools frequently.
(1) Mathematics
(2) Graph
(3) Unit
Unfortunately mathematics is very important although it's hard to study. So many physical laws are expressed by mathematical formulas. We can not escape from math to study physics.
And Drawing graph is a also good tool for physics. Let's see below graph.
Figure 2-2
We can see the status of speed according to time change. We assumed the driver stepped an accelerator at 5:30 pm. So from that time the speed will increase continuously. We can draw a graph like above, and we can understand the car's motion with the graph.
3rd tool, unit, the physical unit is also very important to describe physical phenomena. As I told you when we see a physical unit we can guess the meaning. Look into the unit of velocity. It's (m/s). It means one meter distance moved per second. So we can understand that the definition of velocity. It's the rate of distance with respect to time. Exactly speaking it's the rate of displacement with respect to time. Distance moved is the same with the magnitude of displacement.
I will describe it by mathematics.
velocity
This time, let's focus on the acceleration. Today's theme is the car's acceleration. However please keep in mind the other explanations.
When we see the car's motion, it becomes faster continuously. Because the accelerator has been stepped softly continuously. The driver doesn't loose it. It's called a constant acceleration motion. The acceleration will not be changed. Its magnitude and direction will not be changed.
We can guess acceleration is related to velocity. Let's check its unit.
m/s2
So, if acceleration is 1m/s2, and initial speed is 20 m/s, then the speed will be 21 (m/s) after 1 second. After 10 second the speed will be 30 m/s.
displacement or distance, velocity or speed, constant acceleration, time, between those 4 physical quantities, there are physical formulas
(1) V = V0 + at
(2) S = V0 t + 1/2 * at2
(3) 2aS = V2 - V02
( t: time to be taken for S distance moving of the car ,
a: constant acceleration ,
S: distance moved after time t ,
V0: initial velocity ,
V: velocity after time t, or velocity when the car moves S distance )
Actually we have to study them deeply with using mathematics. We have to learn the concept of differentiation and integral calculus. But I will skip them. I hope later we can look into them later.
Anyway by stepping on the gas pedal (accelerator), the car got constant acceleration toward (+) direction. It's the same direction with velocity. Therefore the speed will be bigger and bigger continuously.
So we can guess the driver may arrive more earlier for home. Then let's calculate the arrival time.
The car is on (-) 10 (km) position when we set his house by origin (0 point). In other words he have to go 10 km more for his home. And the constant acceleration is 1 m/s2 , the initial velocity is 20 m/s, both of them are (+) direction.
To calculate the arrival time, I will use 3rd formula.
Be careful, to calculate we need to convert unit by SI(mks) unit.
10 km is equal to 10 x 10^3 (m), that is 10 x 1000 (m).
2aS = V2 - V02
2x1x10x10^3 = V2 - (20)2
V= 140 (m/s)
We've calculated the speed of the car when it has arrived at home.
Again with using 1st formula,
V = V0 + at
140 = 20 + 1 * t
t = 120 (s) = 2 x 60 (s) = 2 (minute)
So he will get to home at 5:32 (pm).
HW 2-1) Drive the 3rd formula for constant acceleration motion with using formula 1 and 2.
