Motion in One Dimension

What is Motion?

Change in position of an object.

For example:

•A man is walking. He is changing his position so he is in motion.
•Bus is moving. It is changing its position so it is in motion.
•Earth is revolving around the sun. It means that it is changing its position so it is in motion. 
A body

A certain amount of matter limited in all directions and consequently having a finite size, shape and occupying some definite space is called a body.

Particle

A particle is defined as a portion of matter infinitesimally small in size so that for the purpose of investigation, the distance between its different parts may be neglected. Thus, a particle has only a definite position, but no dimension.

Motion in one dimension

Motion

An object or a body is said to be in motion if its position continuously changes with time with reference to a fixed point (or fixed frame of reference).

Mainly motion of a body can be of three types: rectilinear or translatory motion, circular or rotatory motion and oscillatory or vibratory motion.

Rectilinear motion is that motion in which a particle or point mass body is moving along a straight line.

If a particle or point mass body is moving on a circle then such a motion is called circular motion.

In vibratory motion, a particle moves to and fro or back and forth repeatedly about a fixed point such that the amplitude is very small

Note:  The moving object is either a particle, a point object (such as an electron) or an object that moves like a particle. A body is said to be moving like if every portion of it moves in the same direction and at the same rate.

When the position of object changes on a straight line i.e. motion of object along straight line is called motion in one dimension.

To understand the essential concepts of one dimensional motion we have to go through some basic definitions

FRAME OF REFERENCE

The platform can be seen from a moving train and it appears that all  objects placed on the platform are constantly changing their position. But the person on the pier concludes that the objects on the pier are at rest. This means that if we take the trains as the reference frame, the objects are not in place, and if the reference frame is taken as the platform, the objects are in place. Thus, the study of movement is a combined property of the object being studied and the observer. Therefore, it is necessary to define a frame of reference based on which we must study the movement of the object.  A frame of reference is a set of coordinate axes in relation to which position or motion can be determined, or in relation to which physical laws can be expressed mathematically.

 POSITION OF AN OBJECT 

The position of a particle refers to its position in space at a given moment. It concerns the question - "where is the particle at a given moment?" The position of the object is defined relative to the reference frame. By convention, we define the location of a point (essentially treating an object as a point mass) using three coordinates X, Y, and Z. Thus, X, Y, Z is a set of coordinate axes that represent a 3-dimensional space. , and each point in that space can be uniquely defined by X, Y, and a set of Z coordinates, with all three axes perpendicular to each other. Each other A line drawn from the origin to a point represents the position vector of that point.

 Position vector

It describes the instantaneous position of a particle with respect to the chosen frame of reference. It is a vector joining the origin to the particle. If at any time, (x, y, z) be the Cartesian coordinates of the particle then its position vector is given by vector  r= xi^ + y j^+ zk^.

In one-dimensional motion: vector r = x i^,     y = z = 0   (along x-axis)

In two-dimensional motion: vector r= xi^ + yj^  (in x-y plane z = 0)

        In the figure above, the position of a point P is specified and vector  OP  is called the position vector.

 Motion in One Dimension

One dimension implies motion along a straight line or in a single direction. Consider a car or a person driving down a straight road or jogging on a straight track. Think of an object being tossed vertically into the air and then watching it fall. These are examples of one-dimensional motion. There are four main factors to keep track of when evaluating the motion of objects. Time, displacement, velocity, and acceleration are the four variables; time is a scalar, and the other three are vectors.

 Distance and Displacement

Distance travelled or path length is the actual distance covered by the moving object in a given interval of time. It is a scalar quantity.

The change in the position of the object along a particular direction is called displacement.

Displacement Δx = xf – x0

xf is the final position of the object.

x0 is the initial position of the object.

The difference between your ending position (xf) and your beginning point (xo) is known as displacement:

Displacement is a vector quantity. This means it has direction and magnitude, and it is visually depicted as an arrow pointing from the starting point to the ending location. Your displacement is 5 metres north if you start in a specific location and then move north 5 metres from where you started. Then, if you turn around and return with a 5 m south displacement, you will have travelled a total distance of 10 m, but your net displacement will be zero because you are back where you began.

Speed and Velocity

The time rate of change of the position of the particle is called speed. Speed tells about how fast or how slow the particle is moving. It gives the distance covered by the particle in unit time. Speed is a scalar quantity, and its SI unit is m/s. The speed of the object will never be negative; it will either be positive or zero.

The time rate of change of position of an object in a particular direction is called velocity. Velocity can be defined as the time rate of change of displacement. Speed in a particular direction can be called velocity. Velocity is a vector quantity with both direction and magnitude. The unit of velocity is m/s. The value of velocity can be positive, negative or zero.

Average Speed and Average Velocity

The average speed of motion of a particle is defined as the ratio of the time travelled to the elapsed time.

Average speed = total path length/total time taken

Average velocity is defined as the total displacement by the body in time t.

Average velocity = total displacement/total time

Vavg = (Δx/Δt) = (xf – x0)/(tf – t0)

Vavg = average velocity

Δx = change in position

Δt = change in time

xf and x0 are the beginning and ending positions at time tf and t0, respectively.

Instantaneous Velocity

If we consider an infinitesimally small time interval, we get more detailed information. The average velocity becomes the instantaneous velocity for an infinitesimally small time interval of time or the velocity at a single moment over such a time range. A car’s speedometer displays the instantaneous speed. However, for calculating the time taken to travel from one point to another, average velocity is needed. The average velocity at a specific point in time or across an infinitesimally brief time interval is called instantaneous velocity. The instantaneous velocity can be found by taking limits.

Uniform Velocity

A body is said to be moving with uniform velocity if equal changes of displacement take place in equal intervals of time, how small these intervals of time may be. When the body moves with uniform velocity, neither the magnitude nor the direction of velocity changes.

Acceleration

Any process in which the velocity varies is referred to as acceleration. As velocity involves both speed and direction, a body gets accelerated when there is a change in speed, direction or both. If there is no change in speed or direction of the object, there will not be any acceleration, no matter how quickly the object moves. A jet moving at 800 miles per hour along a straight line has zero acceleration, despite the fact that it is moving very quickly. It will have acceleration while it lands because the jet is slowing down.

Average acceleration is defined as the rate of change of velocity with time.

a = (Δv/Δt)= (vf – vi)/Δt

v= initial velocity

vf = final velocity

Δt = time

The SI unit of acceleration is m/s2.

The instantaneous acceleration is

Uniform or Constant Acceleration

A body is said to be moving with uniform or constant acceleration if its velocity changes by equal amounts in equal intervals of time, how small these time intervals may be. When the body moves with constant acceleration, the average and instantaneous acceleration of the body are equal.

Relative Velocity

Imagine yourself standing by the side of a road observing two cars: A and B, both moving towards right with a speed of 10 m/s. Let us consider car A for a moment. According to you its velocity is + 10 m/s. But an observer in car B will measure the speed of car A as 0 m/s. His observations about the velocities are different. According to him, car A is at rest and the road itself is moving with -10 m/s.

Hence whenever we speak about the velocity of any moving object, we must also specify the reference from where we are observing it.

Velocities observed from the ground will be called as velocities relative to the ground. Velocities observed from the car B will be called as velocities relative to the car B.

            Hence, velocity of car A relative to the ground = 10 m/s.

                        Velocity of car A relative to the car B = 0 m/s.

In symbolic notation we write:

             VAG  = velocity of A relative to the ground (G) = 10 m/s.

              VAB  = velocity of A relative to the car B = 0 m/s.

In general we have the relation:

                VAB =  VAG -   VBG

 Note: When specifying the velocities relative to the ground, we usually write:

(i)      VA and  VB  instead of  VAG  and  VAG  . In that case the formula for the relative velocity should also be remembered as:

             vAB =    vA - vB

i.e velocity of A relative to B =

velocity of A (relative to the ground) – velocity of B (relative to the ground)

(ii)       The formula:

              VABVA - VB

Involves vectors and hence velocities should be added or subtracted like vectors while using this relation.

 Example:      A man swims at an angle  to the direction of water flow with a speed vMW = 5 km/hr relative to water. If the speed of water vw = 3 km/hr, find the speed of the man with respect to ground.

Solution:            

                                          

Kinematics Equation when the Acceleration is Constant

A particle moving along a straight line with constant acceleration is said to execute uniformly accelerated motion in one dimension. This type of motion is the simplest kind of accelerated motion. The velocity of the particle changes at the same rate throughout the motion. Some simple equations which relate displacement (S), initial velocity (u), final velocity (v), time taken (t), and uniform acceleration (a) can be obtained. The equations are called kinetic equations for the one-dimensional motion of the particle. The equations are

1. v = u + at

v = final velocity

u = initial velocity

a = uniform acceleration

t = time taken

2. S = ut + (½)at2

S = displacement

u = initial velocity

t = time taken

a = uniform acceleration

3. v2 = u2 + 2aS

v = final velocity

u = initial velocity

a = uniform acceleration

S = displacement

4. Sn = u + (a/2)(2n – 1)

S= displacement in the nth second of motion.

5. a = (S2 – S1)/n2

S2 and Sare the displacements in two consecutive equal intervals of time ‘n’.

Graphs of Straight line motion

With the help of graphs we visualize the variation of position(x), velocity(v),  and acceleration (a)  of a moving particle with time. Plotting time(t) on X-axis and x,v,a on Y-axis we get three useful graphs:

(i)     x-t graphs         (ii)   v-t graphs      (iii) a-t graphs

Note: These graphs will have validity only if motion under study is along a straight line. Then, displacement, velocity and acceleration vectors are collinear and can be treated as algebraic quantities.

 Position - Time Graph           

If we plot time t along the x-axis and the corresponding position (say x) from the origin O on the y-axis, we get a graph which is called the position-time graph. This graph is very convenient to analyse different aspects of motion of a particle. Let us consider the following cases.

(i)     In this case, position (x) remains constant but time changes. This indicates that the particle is stationary in the given reference frame. Hence, the straight line nature of position-time graph parallel to the time axis represents the state of rest. Note that its slope (tan θ) is zero.

                                         

(ii)    When the x-t graph is a straight line inclined at some angle with the time axis, the particle traverses equal displacement Δx in equal intervals of time Δt. The motion of the particle is said to be uniform rectilinear motion. The slope of the line measured by Δx/Δt = tanθ represents the uniform velocity of the particle.

                                     

(iii)    When the x-t graph is a curve, motion is not uniform. It either speeds up or slows down depending upon whether the slope (tan θ) successively increases or decreases with time. As shown in the figure the motion speeds up from t = 0 to t=t1 (since the slope tan θ increases). From t=t1 to t=t2, AB represents a straight line indicating uniform motion. From t=t2 to t=t3, the motion slows down and for t>t3 the particle remains at rest in the reference frame.

 

Example:      The adjacent figure shows the displacement-time graph of a particle moving on the x-axis. Choose the correct option given below.

 

  1. The particle is continuously going in positive x direction.
  2. The particle is at rest.
  3. The particle moves at a constant velocity upto a time t0, and then stops.
  4. The particle moves at a constant acceleration upto a time t0, and then stops.

Solution:      (C) Up to time t0, particle is said to have uniform rectilinear motion and after that comes to rest as the slope is zero.

  The Velocity - Time Graph

The velocity-time graph gives three types of information

  1. The instantaneous velocity.
  2. The slope of the tangent to the curve at any point gives instantaneous acceleration.

              a = dv/dt = tan θ

  1. The area under the curve gives total displacement of the particle.

 

Let us consider the uniform acceleration. The velocity-time graph will be a straight line.

The acceleration of the object is the slope of the line CD.

        a = tan θ = BC/BD = (v-u)/t

        v = u + at                                                         (1)

        The total displacement of the object is area OABCD

        s = Area OABCD = OABD + Δ BCD

        s = ut + (1/2) at2                                                    (2)

        Again

                s = Area OABCD

                = (½)(AC + OD) x OA = (1/2)(v + u)xt

 

Example:      From the velocity-time plot shown in figure, find

(a) Distance travelled by the particle during the first 40 seconds.

(b) Displacement travelled by the particle during the first 40  seconds.

(c) Also find the average velocity during the period.

Solution:      (a)    Distance = area under the curve

                     = 1/2 x 20 x 5 + 1/2 x 5 x 20=100m

                                

                        For distance measurement, the curve is plotted as in Fig.

(b)    Displacement = area under the curve = 0        

(c)    Vav = Displacement/time

                        As displacement is zero,

                        .•.      Vav = 0

 

Example:      The velocity-time graph of a moving object is given in the figure. Find the maximum acceleration of the body and distance travelled by the body in the interval of time in which this acceleration exists.

                       

Solution:      Acceleration is maximum when slope is maximum

                     amax = (80-20)/(40-30) = 6m/s2

                     S = 20 m/s x 10 s + 1/2 x 6 m/s2 x 100s2 = 500m

 

The acceleration-time graph

Acceleration time curves give information about the variation of acceleration with time. Area under the acceleration time curve gives the change in velocity of the particle in the given time interval.

  

 

 Analysis of Uniformly Accelerated Motion

Case-I:
For uniformly accelerated motion with initial velocity u and initial position x0.                                          

 

 Velocity time graph

       

In every case tanθ = a0

 Position time graph

       

Initial position x of the body in every case is x0 (> 0)

Case II:

For uniformly retarded motion with initial velocity u and initial position x0.

                                        

 Velocity time graph

    

In every case tanθ  = -a0

 Position time graph

 

Initial position x of the body in every case is x0 (> 0)

 

Example:      A particle is moving rectilinearly with a time varying acceleration a = 4 - 2t, where a is in m/s2 and t is in sec. If the particle is starting its motion with a velocity of -3 m/s from x = 0. Draw a-t, v-t and x-t curve for the particle.

Solution: