听与学科学/直线运动
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Linear means, straight line. Movement in a straight line is called, linear motion.
We are aware that, everything can be mapped, in four basic directions. North, East, West and South. The word news, is derived, from the first character, of the four directions. North, East, West and South. N E W S. News. We are aware that, the atlas of the world is mapped, according to longitudes, and latitudes. Longitudes are imaginary lines, in the north south axis. Latitudes are imaginary lines, in the east west axis. Any place in the planet, can be identified with its, coordinates. For example, Banglore is 12 degree 58 minutes north, and 77 degree 34 minutes east. This defines the exact position, of banglore in the planet. In fact, the GPS in the mobile phone can exactly pin point, your location, and coordinates.
When there is movement, it is sometimes useful to know, in which direction, the movement is. We are used to say, turn right, go straight, turn left etc. We can also say, we are travelling, north to Delhi, east to Japan, west to U S, or south to Srilanka. Sea navigation, air navigation, is fully dependent on, accurately knowing positions, and directions. Today, even road navigation using GPS, uses positions and directions. When something moves, it moves from a particular position, in a particular direction, to another position.
Graph, is a useful way, to represent many things. We can represent position also, on a graph sheet. Graph sheet has, an X axis and a Y axis. The X axis and the Y axis, intersect in the centre. This position is donated as, 0,0. Other positions can be visualised, relatively to the centre, of the graph. We have an X axis to the right, which we can term as, the East direction. We have an X axis to the left, which we can term as, the West direction. We have a Y axis in the top, which we can term as, the North direction. We have a Y axis to the bottom,, which we can term as, the South direction. The X axis on the left is also called, the negative X axis. The Y axis at the bottom is also called, the negative Y axis. As an example, we can use the graph sheet with a suitable scale, like 1 centimetre equals to, 1 kilo meter. Let us say, we want to plot a point "P", 3 kilo meters east, and 4 kilo meters north. This can be done by positioning the point, 3 centimetres on the X axis, and 4 centimetres on the Y axis. The point we plot, has the co-ordinates, 3 comma 4. The number 3 is the, X co-ordinate. The number 4 is the, Y co-ordinate. So, the point "P" has, the X, Y co-ordinates, 3 comma 4.
Displacement is a measure, to the change in position. It is the shortest distance, between two points, A and B. The shortest distance between 2 points, is always a straight line, or linear.
Distance, is the length of the path, travelled from Point A, to Point B.
Let us imagine you are standing at, your school entrance. You walk straight, to the school gate, which is 25 meters away. You turn and walk back 25 meters, to the school entrance. You have walked a total distance, of 25 plus 25 which is, 50 meters. But, your displacement is 0. This is because, you are in the same place, that you started, and your co-ordinates have not changed. Let us take, another example. Let us assume your school has a compound wall, 100 meters in length, and 50 meters in width. You start walking at the gate. You walk all around the compound wall, and return to the gate. The total distance you have walked, will be 50 plus 100, plus 50, plus 100 meters. That is, you have covered a distance of, 300 meters. But your displacement, is zero. This is because, you are back at the school gate, and your position, has not changed. This is the difference between, displacement and distance. In this module, we will discuss, "Distance".
Speed is the rate, at which, distance is travelled. Let us take the example of walking, from the school entrance, to the gate and back. We covered a total distance of 25 plus 25, which is 50 meters. Let us say, we took 25 seconds, to cover this distance. The speed at which we walked, will be 50 divided by 25, which is 2 meters, every second. We can say that, we walked at a speed of, 2 meters per second.
Let us take the other example, of walking along the compound wall. We covered a total distance, 50 plus 100, plus 50, plus 100 meters. This was a total of, 300 meters. If we take 150 seconds to walk this distance, the speed will be, 2 meters per second. Let us say, we take a bicycle, and cover the same distance in 30 seconds. Now, the speed of the bicycle will be, 300 divided by 30, which is 10 meters per second. To summarise, we can say, our walking speed was, 2 meters per second. The Bicycle speed was, 10 meters per second.
Speed can be expressed in, a simple formula. Speed =, distance travelled, divided by time taken. In S I units, length is represented by meters, and time is represented, by seconds. In S I units, speed is represented by meters per second. The dimensions of speed is, length divided by time. The same formula, can be expressed as, Distance =, speed into time.
To take the example, of walking to the school gate and back, Speed was, 2 meters per second, and time was, 25 seconds. Distance is equal to, 2 into 25, which is equal to, 50 meters.
To take the example, of bicycling along the compound wall, Speed was, 10 meters per second. Time was, 30 seconds. Distance covered is, 10 into 30, which is equal to 300 meters.
We can also express, the speed equation as, Time =, distance divided by speed. In the school gate example, Distance equal to, 50 meters. Speed is equal to, 2 meters per second. Time is equal to, 50 divided by 2, which is equal to 25 seconds.
Let us take the example of, bicycling around the compound wall. Distance equal to, 300 meters. Speed is equal to, 10 meters per second. Time is equal to, 300 divided by 10, which is equal to, 30 seconds.
Ways in which, we can use the speed formula.
In the simple formula, speed is equal to distance, divided by time. The three variables are, Speed, Distance and Time. If we know any two of them, we can calculate, the third variable. If we know distance and time, we can calculate, Speed. If we know speed and time, we can calculate, distance. If we know distance and speed, we can calculate, Time.
Motion can be represented, in a graph. Let us take the example, of walking, at 2 m/sec. We will use the X axis, to represent time, in seconds. We will use the Y axis, to represent distance in meters. If we walk at a speed of 2 m/sec. After one second, we would have reached a, distance of 2 m. We can plot this point, on the graph. X co-ordinate, is one second. Y co-ordinate, is 2 m. After 2 seconds, distance = 4 m. After 3 seconds, distance = 6 m. After 5 seconds, distance = 10 m. After 10 seconds, distance = 20 m. All these points can be plotted, in the graph. All these points can be connected, with a straight line. When speed is uniform, the line joining the points, will be a straight line. It will be inclined to, the X axis. If we take the last point. Distance = 20 m. Time = 10 sec. We can divide the Y co-ordinate, by the X co-ordinate. The Y co-ordinate, represents the distance = 20 m. The X co-ordinate, represent the time = 10 sec. Speed, is Y, divided by X =. 2 m/sec. In this way, we are able to represent motion in a graph. And derive the speed from it.
We can also plot, time verses speed, in the graph. X axis represents, time in seconds. Y axis represents, speed in meters per second. Since the speed is uniform, the Y co-ordinates will always be, 2 m/sec. So, the plotted line will be a straight line, parallel to the X axis. Uniform speeds, when plotted will always be a line, parallel to the X axis.
Many times, a problem has to be formulated, in a way, which can be, easily solved. Let us discuss an example. Your mother left the house, to go to a shop. She walks, at a speed of 2 meters per second. After she left, you realise that, she has forgotten, to take her purse. What should you do? Call her on the mobile? This is not possible, because, the Mobile is in the purse. So you decide to go, and give the purse to her. Now it is already 100 sec, since she left. You decide to go by bicycle, to catch up with her, because, you can cycle at 12 m/sec. When will you catch up with her? Let us formulate this problem. Let us assume that, you will catch up after, "T" seconds. In "T" seconds, you can travel, 12 multiplied by T =. 12 T meter. In the 100 sec, your mother would have covered, 100 multiplied by 2 =. 200 m. In the T seconds, you take to catch up, she would have covered, 2 T meters. Since, you will catch up with her after T seconds, we can equate the distance she would have covered, and what you covered. So 200 + 2 T =. 12 T meter. Transposing 12 T minus 2 T =. 200. 10 T =. 200. T =. 20 sec. You will catch up with her in 20 sec. In 20 sec, she would have covered 240 m. In 20 sec, you would have covered 240 m. So, both of you will definitely meet. This is an example of, how seemingly bigger problems, can be broken down, into smaller problems. These smaller problems can be solved, with the simple speed formula.
The S I unit of speed, is meter per second. For convenience, speed can also be expressed, in kilometres per hour. Note the dimension of speed, remains the same. The dimension is still, length divided by time. For example, we will take a bus, which is travelling at, 40 km/hr. If you travel in this bus for 15 min, which is quarter hour or .25 hrs, you will cover 10 km. That is, distance =. speed multiplied into time. Which is .25 multiplied by 40 =. 10 km. This might be the distance to your home, from the school.
Let us take a train, which is travelling at 80 km/hr. In 5 hr this train will cover 80 multiplied by 5 =. 400 km. This is about the distance between, Banglore and Chennai.
If we take a jet aircraft, which is travelling at, 800 km/hr. In 2 and a half hours, the jet would have covered, 2.5 multiplied by 800 =. 2000 km. This is about the distance, from Banglore to Delhi.
Sound travels at a speed of about, 1200 km/hr. When we are near hills or mountains, if we make a loud noise, we hear an echo. The echo is nothing but, the time taken for sound to travel to the hill, and bounce back to our ears. If the hill is closer, the echo will be faster. If the hill is far, the echo will take longer. There are many applications, which use this simple principle. A few examples. A submarine was designed to travel, under the sea, to escape detection. We can send sound waves, to the submarine, which will bounce back. If we measure the time taken, for a sound wave to come back, we can calculate, the distance of the submarine. This is done by using, the speed of sound, and the simple speed formula.
Echo cardiograms are used by doctors, to see your heart. Do not worry, they will not cut you open, to do this. By bouncing sound waves, off the heart, the echo cardiogram, is able to build an image of the heart. It uses the same speed formula, that we have been using so far.
Bats are birds that live in, dark caves. They cannot see very well. When they fly, they make clicking sounds. By listening to the echo, of their own clicks, they are able to navigate, even in the dark. This is called, echo location.
Light travels at a speed of about, 1 billion kilo meter hour. Radio waves also travel, at the speed of light. We all know about radar. Radar transmits radio signals, in the air. This signals bounce off aircraft, which could be travelling very far away. Listening to the echo of the radio wave, Radar is able to measure the time taken, for the signal to return. With this time, Radar is able to calculate, the exact distance of the aircraft. Again, they use the same speed formula, and speed of radio waves.
Laser beams are concentrated beams, of light. By bouncing the laser beam to the moon and back, scientists, calculate the distance to the moon.
We might think all these equipments, are very sophisticated, and expensive. Architects, used to carry a measuring tape, and take measurements, with an assistant, and may be a ladder. Architects, now carry, a small match box sized measuring instrument, to make all measurements. This small inexpensive instrument uses the principle, of bouncing off light and measuring, the time taken to return.
G P S stands for, Global Positioning System, Satellites can receive, and transmit radio waves. Radio waves can be received, on earth in any location. By using the same principle, of speed of radio waves, G P S systems are built. With a G P S phone, we can locate anybody, anywhere on the planet.
The light from the sun, takes about 8 min and 10 sec, to reach the earth. After this discussion, it will be quite easy to calculate, the distance of the sun.
You want to think further than that? A light year is the amount of distance, travelled by light, in one year. Our nearest galaxy, is the Andromeda galaxy. The Andromeda galaxy is 2.5 million, light years away. Our humble speed formula can be used, to calculate the distance, to this galaxy.
A car is travelling a total distance of, 100 km. Part of the time, it is travelling at, a slow speed. Part of the time, it is travelling at, a high speed. The total time of travel, is 2 hrs. In this case, we can calculate the average speed, of the car. The average speed is =, total distance travelled, divided by time taken. This is =, 100 km divided by 2 hrs. The average speed =, 50 km/hr.
Velocity specifies both the magnitude, and direction of motion. For example, a car is moving with a velocity of, 50 km/hr in the east direction. We call velocity, as a "Vector". Vector has both magnitude, like 50 km/hr, and direction like East. When we specify speed, we give only the magnitude. For example, a car is moving at 50 km/hr. We call this as, "Scalar". A scalar has magnitude, but no direction. 50 km/hr East, is a vector, which is velocity. 50 km/hr is a scalar, which is speed. If an object is moving in the same direction, magnitude of the velocity, and the speed, will be the same. In this module, we will only discuss objects in Linear Motion, moving in the same direction. So, in this module, magnitude of velocity, will be same as speed.
Acceleration is the rate of change of velocity. We will discuss only, uniform acceleration. Let us say, the initial velocity of an object, is "U" meters per second. The object keeps increasing its velocity. It reaches a final velocity, of "V" meters per second. It takes "T" seconds to reach, a final velocity of "V" meters per second. Acceleration = final velocity minus, initial velocity, divided by time. The S I unit of acceleration, is meters, divided by second squared. The dimension of acceleration, is length, divided by time squared.
You start your bicycle, from a stationary state. You pedal your bicycle, to a final velocity of 12 m/sec, in 12 sec. Acceleration = final velocity minus, initial velocity, divided by time. That is, acceleration = 12 minus 0, divided by 12. = 12 divided by 12. = 1 m divided by second squared. Your acceleration was 1 m, for every second you travelled. Your velocity, at the end of 1 sec, is 1 m/sec. Your velocity, at the end of 2 sec, is 2 m/sec. Your velocity, at the end of 3 sec, is 3 m/sec. Your velocity, at the end of 5 sec, is 5 m/sec. Your velocity, at the end of 10 sec, is 10 m/sec. Your velocity, at the end of 12 sec, is 12 m/sec.
When the rate of change of velocity is constant, the acceleration is called, uniform acceleration. In the example just discussed, the velocity changes by 1 m/sec, every second. The rate of change of velocity, is a uniform 1 m/sec, every second. This is an example of uniform acceleration. Average velocity =, initial velocity + final velocity, whole divided by 2. Average velocity =, "U" + "V", whole divided by 2. In the bicycle example we just discussed, Initial velocity =, 0. Final velocity =, 12 m/sec. Average velocity =, 0 + 12, whole divided by 2. Average velocity =, 6 m/sec.
An object starts moving, with an initial velocity of "U". It accelerates at a rate, of "A". It reaches a final velocity, of "V". It takes a time "T", to do this. We know that, acceleration = final velocity minus, initial velocity, divided by time. That is, "A", = "V" minus "U", divided by "T". Transposing, we get "V" minus "U" = "A", multiplied by "T". Transposing again, "V" = "U" + "A T". Final velocity = initial velocity + acceleration, multiplied by Time. We give this formula a pet name, VUTA formula. Beware that, this is only a pet name, and not officially used anywhere. This formula deals with 4 variables, final velocity, initial velocity, acceleration, and time. If we know any three variables, we can calculate the fourth. If we know "U", "A", and "T", we can calculate "V". If we know "T", "A", and "V", we can calculate "U". If we know "A", "V" and "U", we can calculate "T". If we know "V", "U" and "T", we can calculate "A".
Example.
"A" car starts from initial velocity of 40 km/hr. It has an acceleration of, 10 km/hr squared. It accelerates for, 1 hr. Using "V" = "U" + "A T". "V" = 40 + 10 multiplied by, 1. Final velocity "V", = 50 km/hr.
We can also see that. "U" = "V" minus "A T". "U" = 50 minus, 10 multiplied by 1. "U" = 50 minus 10. Initial velocity "U" = 40 km/hr.
We can also see that. "T" = "V" minus "U", divided by "A". "T" = 50 minus 40, divided by 10. "T" = 10, divided by 10. "T" = 1 hr.
We can also see that. "A" = "V" minus "U", divided by "T". "A" = 50 minus 40, divided by 1. Acceleration "A" = 10 km, per hour squared.
This formula deals with 4 variables, distance travelled, initial velocity, acceleration and time. Distance is denoted by "S". "S" = "U T" + half "A, T" squared. This can be easily derived. Distance travelled = average velocity, multiplied by time. Average velocity = "U" + "V", whole divided by 2. Distance travelled = "U" + "V", whole divided by 2, whole multiplied by "T". Substituting "V" = "U" + "A T". "S" = "U" + "U" + “A T" , divided by 2, multiplied by "T". Expanding, "S" = 2"U T", divided by 2, + "A T" squared, divided by 2. So, "S" = "U T" + half "A, T" squared. We will give this formula, the pet name as SUTA formula. It deals with 4 variables, distance, initial velocity, acceleration and time. If we know any three variables, we can calculate the fourth.
Example.
Taking the car example. Initial velocity "U" = 40 km/hr. Acceleration = 10 km/hr. Time = 1 hr. "S" = "U T" + half "A, T" squared. "S" = 40 multiplied by 1, + half multiplied by 10, multiplied by ,1 squared. "S" = 40 + 5, = 45 km.
This formula deals with 4 variables, Final velocity, initial velocity, acceleration and distance travelled. "V" squared =, "U" squared + 2 "A, S". This can be easily derived. "V" =, "U" + "A T". Squaring both the sides. "V" squared =, "U" + "A, T" whole squared. "V" squared =, "U" squared, + "A" squared, "T" squared, + 2 "U A T". "V" squared =, "U" squared, + 2 "A" multiplied by, whole of "U T" + half “A T" squared. We know that, "U T" + half “A T" squared, = "S". Therefore. "V" squared =, "U" squared + 2 "A S". We will give this formula, the pet name as VUSA formula. It deals with 4 variables, Final velocity, Initial velocity, distance and acceleration. If we know any three variables, we can calculate the fourth.
Example.
In the car example. "V" squared =, "U" squared + 2 "A S". "V" squared =, 40 squared, + 2 multiplied by 10, multiplied by 45. "V" squared =, 1600 + 900. "V" squared =, 2500. Taking the root. "V" =, root of 2500. "V" =, 50. Final velocity is 50 km/hr.
Using these 3 simple formulas, most of the problems concerning linear motion, can be solved. To summarise these formulas.
注意：名称仅为宠物名称，仅在本模块中使用。
VUTA. "V" =, "U"+ "A T". SUTA. "S" =, "U T" + half "A T" squared. VUSA. "V" squared =, "U" squared + 2 "A S".