﻿ One D motion

Kinematics: One dimensional motion

Kinematics is studying the motion of objects without considering the causes (forces) of the motion. One dimensional (1-D) kinematics or motion refers to the motion of objects on straight line paths. When studying one dimensional motion, we do not consider the size or the shape of the object, instead we consider it as a point. The positions of the point at different times lie on a straight line in a 1-D motion. Examples of 1-D motion: a car moving on a straight road, motion of a dropped object or the motion of a foot ball kicked vertically upwards.

Frame of reference

We use the Cartesian coordinate system with three perpendicular axes, $x$, $y$, and $z$ as reference, and do measurements with respect to that. We call this reference, frame of reference or the reference frame.

Since in a one dimensional motion, the object moves on a straight line path, we consider that path is on one of the axes, usually the $x$ or the $y$ axis.

In a 1-D motion, an object can have only two possible directions. If we take the path of the object is along the $x$ axis, then the object can move in the positive direction (i.e., $+x$) or in the negative direction ($-x$). Taking a direction as positive or negative is arbitrary. If you take the east as positive, then the west will be the negative direction and vice versa.

Position of an object

We use the letter $x$ or $y$ to represent the position of the object in a 1-D motion. If we consider the path is on the $x$ axis, then we use the letter $x$ and if the $y$ axis is the path, then we use $y$ for the object's position.

Distance and displacement

Distance and displacements are two different things in physics.

If an object moved from a point A to another point B, the distance traveled by the object is the length of the path from A to B. There may be many available paths from A to B, of different lengths. In the figure below, there are two possible paths from A to B, a shorter path, path 1 and a longer path, path 2. If the object takes path 2, then it will travel more distance than on path 1. So, the distance depends on the path taken by the object. Since distance is a length, it is a positive number. Further, distance has no direction as the distance traveled from A to B is same as the distance from B to A on the same path. A quantity that has no direction is called a scalar, so distance is a scalar.

Displacement of an object is a measure of how much the object has moved or displaced from its starting point and in what direction. To find the displacement, draw a straight line with an arrow, from the initial point to the final point (see the figure above). The length of this straight line is the size of the displacement, called the magnitude of the displacement. And the arrow that points away from the initial position and towards the final position of the object, shows the direction of the displacement. So, displacement has a magnitude and a direction. Any quantity that has both magnitude and direction is called a vector, so displacement is a vector. The displacement line (the straight line with the arrow) is called the displacement vector.

Displacement is just the change in position of the object and is independent of the path as you only need the initial and the final position of the object to find its displacement.

If an object makes a round trip motion and come back to the starting point, then the displacement of the object is zero at the end of the motion. But the distance will not be zero. The distance is just the length of the loop if the object makes one round trip around the loop. But if it makes two round trips, the distance will be twice the length.

Displacement in a 1-D motion

In a 1-D motion, we can define the displacement as

displacement, $\Delta x=x_2-x_1.$

where $x_1$ is the initial position of the object and $x_2$ is its final position.

The displacement, $\Delta x$ is positive for an object moving in the positive direction as $x_2\gt x_1$ in the positive direction. And $\Delta x$ is negative for the negative direction as in that direction $x_2 \lt x_1$. So, in a 1-D motion, the sign of the displacement reflects the direction of the displacement.

Note that you cannot compare distance and displacement as displacement has direction but distance has not. But you can compare the distance and the magnitude of displacement as both represent a length. Except the motion is one dimensional and in one direction, magnitude of displacement is smaller than the distance traveled by the object.

Example: A car travels 5 km due east on a straight road, then turns back and travels 3 km due west. Find the distance traveled by the car and its displacement.

Solution:

Distance travelled by the car is the length of the path covered by the car. That is

distance $= 5 km + 3 km=8 km$.

To find the displacement, put the frame of reference, and take the object's path is along the $x$ axis. Also, consider the object starts from the origin (you can choose another value too).

I took the east as the positive direction, so the west is the negative direction. Since choosing a direction as positive or negative is arbitrary, it doesn't matter if you take west as positive direction and the east as negative.

The initial position of the object is shown as a open circle and the final position as filled circle. Draw a line with an arrow from the initial to the final position, which is the displacement vector.

The arrow of the displacement vector points to the positive direction that is the east. So, the direction of the displacement is eastward.

Writing the initial and the final position of the object,

$x_1=0$ and $x_2= 2 km$.

So, the object's displacement is

$\Delta x = x_2-x_1= 2 km -0 = 2km$.

The displacement is a positive number, so, the direction of the displacement is positive. Since we took positive direction as the east, the direction of the displacement is eastward. This is what we already got from the arrow of the displacement vector.

Absolute value of the displacement is the magnitude of the displacement, which is 2 km.

Distance traveled by the object $= 8 km$.

Displacement of the object is 2 km eastward (write the magnitude first, then the direction).

Speed

To find the speed of a moving object, you measure the distance the object travels over a time period, and divide that distance by the time. So, speed of an object is defined as

$speed=\dfrac{distance}{time}$

Speed can be instantaneous or average. Instantaneous speed is the speed at an instant of time. The speed of your car that you find by looking at the odometer is the instantaneous speed as it tells you the speed of the car at that moment. Average speed is the speed that you determine by observing the motion of the car over a time period (not one instant). For example, assume you drive your car for, say 10 minutes and travel a distance of 2 km. Dividing the distance by time, you get, 0.2 km/min. This speed is the average speed as it is not the speed at one time, but it is the speed determined over a period of time (10 min.).

Why we use the word average for the speed that is measured over a time period ? Because, during the time period, the object can have different instantaneous speeds at different times, so what you will get is the average speed. If the object moves at a constant speed at any moment during the time period, then the average speed will be equal to the instantaneous speed. Further, if the speed is instantaneous, then we skip the word instantaneous and just say speed.

Now, another question may arise, how can I find the instantaneous speed without having a time period to have some distance. Yes, you cannot travel a distance without spending some time. Actually, the instantaneous speed shown by the odometer is not exactly instantaneous, but it is an average speed over a very small time interval, much smaller than the human response time. So instantaneous speed is actually the average speed over a small time interval. How small the time interval should be? It depends upon how quickly the object speed changes around the time of interest and also the needed accuracy.

Since speed is distance divided by time, and distance is has no direction, so speed has no direction and is always positive.

Velocity

Like speed is related to distance and time, there is another quantity, called velocity that relates displacement and time. i.e.,

$velocity=\dfrac{displacement}{time}$

or

$v=\dfrac{\Delta x}{\Delta t}$

where $v$ is the velocity, and $\Delta x$ is the displacement in a time interval, $\Delta t$.
If $t_1$ is the initial time and $t_2$ is the final time, then $\Delta t=t_2-t_1$. Note that $\Delta t$ is always a positive number as time never decrease ($t_2>t_1)$.

Like speed, velocity can be instantaneous or average. Average velocity is the velocity over a period of time, and instantaneous velocity is that at a single instant of time (practically that is over a small time interval). When writing average velocity, we place a bar above the letter, $v$, i.e., $\bar v$ and the instantaneous velocity as $v$, without the bar. Also, if you just say velocity, which represents the instantaneous velocity.

Velocity is a measure of how the position of an object changes per unit time. If we use the SI units, then the velocity of an object tells us how many meters the object's position changes every second.

Since displacement is a vector, and velocity is displacement over time, velocity is a vector. Direction of velocity is the direction of the object's displacement.

When an object is in motion, it needs some time to change its direction. So, if we take a time interval sufficiently small, then during that time interval we can have the object's motion on a straight path in one direction. And in that small time interval, the magnitude of velocity is same as the speed of the object because for motion in one direction, magnitude of the displacement is same as the distance traveled by the object. Since the velocity is instantaneous in a small time interval, the magnitude of the instantaneous velocity is just the speed of the object.

But the magnitude of average velocity is not always the same as the average speed. If the object travels on a straight line in one direction, then the magnitude of average velocity is same as the average speed. But if the object travels back and forth in a one dimensional motion or the object moves on a curved path, then the magnitude of average velocity will be smaller than the average speed.

Direction of velocity is the direction of motion of the object. i.e., in 1-D motion, the velocity is positive if the object is moving in the positive direction and is negative if the motion is in the negative direction. If you take the north as positive direction and the object is moving due south, then the velocity will be negative.

You can show that the velocity of an object is the slope of the position versus time graph as follows.

In the figure above, the position of a moving object at different times is plotted. Let us consider the motion of the object over a time interval from $t_1$ to $t_2$. $x_1$ and $x_2$ are the positions of the object at $t_1$ and $t_2$ respectively. If you draw a line between the points $(t_1, x_1)$ and $(t_2, x_2)$ and find the slope of the line, you will get

slope $=\dfrac{x_2-x_1}{t_2-t_1}$

$=\dfrac{\Delta x}{\Delta t}$

$=\bar v$

Thus, the slope of the line drawn between any two points corresponding to two times is just the average velocity over the time interval between those times. Now, the question is suppose if you take an instant of time, then how the velocity is related to the slope? If you take only one time, then the slope of the tangent drawn at the point corresponding to the time will be the velocity at that time. So, velocity is just the slope of a position versus time graph.

As an example, an object's position ($x$) versus time, ($t$) of an object is given below. I have selected four points A, B, C and D corresponding to 4 different times. The tangents at the points A, B, C and D are shown. Velocity of the object at a time is the slope of the tangent at the point corresponding to that time. i.e., the velocity of the object at $t=2s$ is the slope of the tangent at A.

Slope of the tangents at A and B is positive; the slope of the tangent at C is zero (as the tangent is a horizontal line). So, the velocities at $t=2s$ and at $t=4s$ are positive, and at $t=7.5s$, the velocity of the object is zero. The slope of the tangent at D is negative. Therefore, at $t=9.5 s$, the velocity is negative. Based on these we can conclude that the object is moving in the positive direction up to the point C and it stops there, then turns back and move in the opposite direction (negative direction).

Acceleration

When the speed of a car is increasing, we say that the car is being accelerated. And if the car slows down, the car is decelerating. In both the cases, the velocity of the car changes. In physics, we say there is an acceleration whenever there is a change in velocity of the object.

Acceleration of an object is defined as

acceleration $=\dfrac{change\:in\:velocity}{time}$
i.e., $a=\dfrac{\Delta v}{\Delta t}$

where $a$ is the acceleration; $\Delta v = v_2 - v_1$ is the change in velocity; $v_1$ is the initial velocity; and $v_2$ is the final velocity; and $\Delta t$ is the time it takes to change the object's velocity from $v_1$ to $v_2$.

Like velocity, acceleration can be average $(\bar a)$ or instantaneous $(a)$.

Acceleration is a measure of how the velocity changes per unit time. In SI units, acceleration tells us how much the velocity changes every second. For example, an acceleration (magnitude) of $7 m/s^2$ means, every second the speed of the object increase or decrease by $7 m/s$.

We can show that in a velocity versus time graph (with time $t$ on the $x$-axis and the velocity, $v$ on the $y$-axis) of an object, the slope of the graph is the acceleration of the object. This is same as showing velocity as a slope in a position versus time graph.

Since velocity is a vector, acceleration is also a vector. In 1-D motion, acceleration and velocity of an object are in the same direction if the object is speeding up, but if the object is slowing down, they are in the opposite directions. We can prove these by considering the following:

Assume an object is moving in the positive direction at some speed, say $3 m/s$. This speed is the instantaneous speed, which is the magnitude of velocity. The object is accelerated for $1 s$ (I take this time for easy calculation) so that the speed at the end of $1 s$ is say $5 m/s$. Since the object is moving in the positive direction, the initial and the final velocities are positive:

$v_1= 3 m/s$ and $v_2= 5 m/s$

So the acceleration of the object is

$a=\dfrac{ v_2-v_1}{\Delta t}$

$= \dfrac{5 m/s-3 m/s}{1 s} = + 2 m/s^2$

Acceleration and the velocity both are positive, so they are in the same direction.

Now, let us consider the object is moving in the negative direction with some speed, say $2 m/s$ and is accelerated to a speed of say $6 m/s$ over a time period of $1 s$. Since the object is moving in the negative direction, the velocities are negative:

$v_1= - 2m/s$ and $v_2= -6 m/s$.

Finding the object's acceleration,

$a=\dfrac{v_2-v_1}{\Delta t}$

$= \dfrac{-6 m/s -(-2 m/s)}{1 s} = - 4 m/s^2$

Here again, the acceleration and the velocity are in the same direction as both are negative.

Now, let us consider the object is moving in the positive direction but slowing down. As the object slows down, its final speed is smaller than the initial speed. Assume the initial speed is say $6m/s$ and the final speed as $3 m/s$. And take $\Delta t = 1 s$. Since the motion is in the positive direction, we have

$v_1 = 6m/s$ and $v_2 = 3m/s$

The acceleration of the object is

$a=\dfrac{3 m/s - 6 m/s}{1 s}=- 3 m/s^2$

The acceleration is negative, and the velocity is positive, so they are in the opposite directions. You will get the same result if you take the object is moving in the negative direction. So, when an object is slowing down, the acceleration and the velocity are in the opposite directions.

It is important to note that the sign of acceleration depends on two things, the sign of the velocity (i.e., the direction of motion) and whether the object is speeding up or slowing down. So, a negative acceleration of an object does not always mean the object is slowing down or a positive acceleration does not always mean the object is speeding up.

1-D kinematic equations

In a 1-D motion, we will mostly deal with constant acceleration. When the acceleration of an object is constant, its average acceleration over a time interval will be same as the instantaneous acceleration at any instant of time during that time interval. It is like taking the average of same numbers.
For constant acceleration,$a$, we can have a set of four equations that relate the time, and the object's displacement, velocity and acceleration. These equations are called 1-D kinematic equations. Derivations of these equations are given in a separate page: deriving the kinematic equations.

Following are the 1-D kinematic equations:

$v=v_0+at$

$\Delta x=\dfrac{1}{2}(v_0+v)t$

$\Delta x=v_0t+\dfrac{1}{2}at^2$

$v^2 = v_0^2 + 2a \Delta x$

In the 1-D kinematic equations, $t=0$ is the initial time, and $t$ is the time interval between the initial and the final time. For example, if $t = 2 s$ means, $2 s$ has passed since the object's motion started or from the time that you consider as the starting time. $v_0$ is the velocity at $t=0$; $\Delta x$ and $v$ are respectively the displacement and velocity at time $t$.

Free fall motion

Isaac Newton is an English physicist, discovered gravity in the 1680s. He found that objects are attracted toward the earth by gravity.That is why if you drop or throw an object, it finally comes down instead of going up. But long before Newton, Galileo Galilei, an Italian astronomer and a physicist, discovered that when there is no air resistance, all objects fall with same acceleration irrespective of their masses. i.e., if you drop a $1 kg$ object and a $10 kg$ object at the same time from same height, both will hit the ground at the same time.

But if you simultaneously drop a feather and a rock from the same height, you can see that they do not reach the ground at the same time. The rock reaches the ground earlier than the feather. This is because of the air resistance. Air resistance dominates the feather due to its larger surface area per unit mass than the rock. But if you take the feather and the rock inside of a long evacuated cylindrical container, and drop them, then both will hit the bottom at the same time.

An object in motion above the earth's surface that is only under the influence of earth's gravity is said to be in free fall. But a moving object in air has air resistance acting on, in addition to gravity. We can ignore the air resistance in most cases except the objects like a feather or a piece of paper. So, ignoring the air resistance, any object that you drop or throw into the air is called a freely falling object and the motion of such an object is called free fall motion. In all cases, that we will deal with in this section, we will ignore the air resistance.

If you drop an object or throw an object exactly vertically upward or downward, you can see that the object follows a straight line with respect to the earth. For example, if you throw a ball vertically upward, it will come back to the exact point from where it was thrown. So the motion of such an object (dropped or thrown vertically upward/downward) is one dimensional. We will focus on the 1-D free fall motion in this section.

Let us see what happens if you throw an object vertically upward. The object speed decreases as it goes up, then, it stops at some point, and comes down. While the object is coming down, its speed increases. Whether the object is going up or coming down, you can find that the acceleration of the object is always downwards. This is because, when the object is going up, the speed decreases so the acceleration is opposite to the direction of motion, which is downward. When the object is coming down, the speed of the object increases so the acceleration is in the same direction as the direction of motion, which is downward.

Acceleration of a freely falling object is called acceleration due to gravity. We will use the letter $g$ to represent the magnitude of the acceleration due to gravity. If you measure the acceleration of a free fall object near the surface of the earth, you will get, $g=9.80 m/s^2.$ This value is constant unless you go too far from the earth's surface.

Since the acceleration due to gravity is constant near the surface of the earth, we will use the 1-D kinematic equations to describe the 1-D free fall motion. Since the motion is vertical (vertical to the earth's surface), we will consider the motion is along the $y$ axis. So we use the variable $y$ instead of $x$ for the object's position and $\Delta y$, as the object's vertical displacement:

$v = v_0 \pm g t$

$\Delta y = v_0 t \pm {1\over 2} g t^2$

$v^2 = v_0^2 \pm 2g \Delta y$

I have replaced the acceleration $a$ with $\pm g$ as $g$ is the magnitude and is always positive, so we need to put the sign for the acceleration direction. The sign of the acceleration depends on how you choose the direction. One way is consider the initial velocity direction as positive ($+y)$. Then, look for, in which way the object is moving initially: upward or downward. If it is upward, then $a=-g$ as acceleration is always downward, so you need to have a minus sign in front of $g$ in the equations; and if the initial velocity is downward, then $a=+g$, so you need a $+$ sign in front of $g$.