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Motion

Motion Synopsis

Synopsis

 

Introduction to Motion

  • A particle is said to be in motion if its position, relative to the surroundings, changes with respect to time. 
  • A body which remains stationary (i.e., no change in position) is said to be at rest.
  • Motion and rest refer to the state of bodies described in relation to its surroundings.



  • The movement of a body is called motion. Rest and motion are relative terms. Thus, a body is said to be in motion when its position changes continuously with respect to a stationary object taken as a reference point.
  • Suppose a body is moving very fast (e.g. train, rocket etc.), then it can be easily seen that it is in motion. Now, suppose a body is moving very slowly (e.g. snail, hour hand of watch etc.), then it is difficult to observe its motion.

 

Scalars and Vectors

Scalars

  • In Physics, we can classify quantities such as velocity, speed, acceleration and position as scalars or vectors.
  • Direction is associated with a vector but not with a scalar.
  • A scalar quantity is a quantity only with magnitude. It is specified completely by a single number, along with a proper unit.
  • Examples: The distance between two points, mass of an object, the temperature of a body and the time at which a certain event happened.
  • Scalars can be added, subtracted, multiplied and divided just as the ordinary numbers.

Vectors 

  • A vector quantity is a quantity which has both magnitude and direction.
  • A vector is specified by giving its magnitude by a number and its direction.
    For example, displacement, velocity, acceleration and force.
 
Distance and Displacement

  • Distance travelled is the actual length of the path covered by a moving object irrespective of the direction in which it is travelling.
  • Displacement is defined as the shortest distance between the initial and final positions of the object along with the direction of its travel.



  • Distance travelled has only magnitude associated with it, whereas displacement is associated with both magnitude and direction.
  • Thus, distance is known as a scalar quantity and displacement is known as a vector quantity. Both the quantities are denoted by s; the only difference being that distance has magnitude and displacement has magnitude and direction.
  • Distance covered by a moving object can never be zero. However, the displacement of that object can be zero.
    o Example: An object starts its motion at a point. After travelling along some path, it returns to the same point. Thus, its initial and final positions are the same. Hence, its displacement is zero.

 
Uniform and Non-uniform Motion
 
  • An object is said to have uniform motion if it travels equal distances in equal intervals of time, no matter how small these time intervals are.
  • If we plot a distance–time graph for uniform motion, then it will be a straight line.



    An object is said to have non-uniform motion if it travels unequal distances in equal intervals of time.

  • If we plot a distance–time graph for non-uniform motion, then it will be a curved line.




  • Examples:
    Uniform motion: A motorcycle travelling with a constant speed of 5 m/s will cover equal distances of 5 m for every 1 s.
    Non-uniform motion: A body dropped from the top of a building is acted upon by gravity. Hence, its distance will increase with each interval of time.
  • Thus, to find out the type of motion, we should draw a distance–time graph. If it is a straight line, then it is uniform motion. If it is a curved line, then it is non-uniform motion.
  • The graph of a freely falling body is a curved line. Motion of a freely falling body is accelerated. Thus, we can say that non-uniform motion is an accelerated motion.
 
Speed and Velocity
Speed
  • Speed is a quantity which tells how fast the body is moving.
  • It is defined as the distance travelled by an object per unit time.

    begin mathsize 12px style Speed equals fraction numerator Distance space travelled over denominator Time space taken end fraction end style

  • If s is the distance travelled in time t, then the speed v is given as

    begin mathsize 12px style straight v equals straight s over straight t end style

  • The SI unit of speed is metre per second and is written as m s−1. The CGS unit of speed is centimetre per second and is written as cm s−1.
  • Speed is a scalar quantity, as only the magnitude is needed to be specified.
 
Instantaneous Speed
 
  • Instantaneous speed is the speed of a particle at a given instant. It is defined as the ratio of the distance travelled in an extremely small interval of time tending to zero.
 
Average Speed

  • The average speed of a body is the total distance travelled by a body divided by the total time taken to cover this distance.


    begin mathsize 12px style Average space speed equals fraction numerator Total space distance space travelled over denominator Total space time space taken end fraction end style  
 
Uniform Speed

  • Uniform speed is possessed by an object when it travels equal distances in equal intervals of time, no matter how small these time intervals may be. 
 
Variable Speed

  • A body has a variable speed if it does not cover equal distances in equal intervals of time.
    Example: The car moving on crowded street have variable speed.
 
Velocity 
 
  • The velocity of a body is defined as the distance travelled by an object per unit time in a given direction.
  • We know that the distance travelled in a given direction is known as displacement.
  • If s is the displacement of the object in time t, then the velocity v is given as
  • The SI unit of velocity is metre per second and is written as m s−1. The CGS unit of velocity is centimetre per second and is written as cm s−1.
  • Velocity is a vector quantity, as both the magnitude and direction are needed to be specified.
  • Thus, the velocity of an object can be changed by changing
    oThe speed of the object.
    oThe direction of the object, keeping its speed constant. 
    oBoth speed and direction of the object. 
 
Instantaneous Velocity

  • It is the velocity of the particle at a given instant. The direction of instantaneous velocity is along the tangent drawn to the curve describing the path at that instant if the body undergoes curvilinear motion.

    begin mathsize 12px style straight v with rightwards arrow on top equals fraction numerator stack increment straight s with rightwards arrow on top over denominator increment straight t end fraction
stack increment straight s with rightwards arrow on top space is space change space in space displacement space in space straight a space small space interval space of space time space increment straight t end style
 
Uniform Velocity

  • Uniform velocity is possessed by an object when it travels in a specified direction in a straight line and covers equal distances in equal intervals of time, no matter how small these time intervals may be.
Average Velocity

  • When the velocity of an object is changing at a uniform rate, then the average velocity is given as the arithmetic mean of the initial and final velocity in a particular time interval.

    begin mathsize 12px style Average space velocity space equals space fraction numerator Initial space velocity space plus space Final space velocity over denominator 2 end fraction end style
  • If u is the initial velocity and v is the final velocity for a given time interval t, then the average velocity begin mathsize 12px style straight v with rightwards arrow on top end style of the object is given as
    begin mathsize 12px style straight v with rightwards arrow on top equals fraction numerator straight u space plus thin space straight v over denominator 2 end fraction end style    
 
Variable Velocity

  • If a body undergoes unequal displacements in equal intervals of time, then the body is said to possess variable velocity. Velocity is said to be variable if there is a change either in its magnitude or in its direction or both.
Acceleration
  • Acceleration of a body is defined as the rate of change of its velocity with time.

    begin mathsize 12px style Acceleration equals fraction numerator Change space in space velocity over denominator Time space taken space for space change end fraction
Acceleration equals fraction numerator Final space velocity minus Initial space velocity over denominator Time space taken space for space change end fraction end style

  • If ‘u’ is the initial velocity, ‘v’ is the final velocity and t is the time taken, then the acceleration ’a’ of the body is given as


    begin mathsize 12px style straight a equals fraction numerator straight v minus straight u over denominator straight t end fraction end style
  • The SI unit of acceleration is metre per second square and is written as m s−2. The CGS unit of acceleration is centimetre per second square and is written as cm s−2.
  • Acceleration is a vector quantity.
  • For an object moving with uniform velocity, the acceleration is zero.
 
Uniform Acceleration

  • A body is said to possess uniform acceleration if it travels in a straight line and its velocity increases by equal amounts in equal intervals of time.

    oExample 1: Motion of a freely falling object.
    oExample 2: Motion of a ball rolling down an inclined plane.
  • The velocity–time graph of a body having uniform acceleration is a straight line.

 
 
Non-Uniform Acceleration
 
  • A body is said to possess non-uniform acceleration if its velocity increases by unequal amounts in equal intervals of time.
    oExample 1: Motion of a vehicle on a crowded city road.
    oExample 2: Motion of a 110 m hurdles sprinter in a race.
 
Retardation
 
  • When the velocity of a body increases with time, it is called acceleration. However, if the velocity decreases, then it is called retardation, deceleration or negative acceleration.
  • Retardation is also measured as change in velocity per time taken for change.


    begin mathsize 12px style Retardation equals fraction numerator Change space in space velocity over denominator Time space taken space for space change end fraction end style
    o Example: When a person opens a parachute, its velocity goes on decreasing.
 
Study of Motion Through Various Graphs 
 
  • If the graph of motion of an object is known, then its speed, distance travelled and acceleration can be calculated.
  • Three kinds of graphs can be plotted.

 
 
Distance-Time graph (or displacement-time graph)

  • The distance–time graph of a body moving at uniform speed is a straight line.

  • Speed is distance travelled per unit time.


    begin mathsize 12px style Speed equals fraction numerator Distance space travelled over denominator Time space taken end fraction end style
  • From the graph, we have

    begin mathsize 12px style Speed equals OC over OB end style
  • But, the slope of the graph = begin mathsize 12px style OC over OB end style
  • Thus, the slope of the distance–time graph gives the speed of the object.
  • If the speed of the object is non-uniform, then the graph will be a curved line.



  • If a displacement–time graph is drawn, then the slope will give the velocity of the object.
 
Speed-Time Graph (or Velocity-Time Graph)
 
  • There are three types of speed–time graphs for a body.


     
 
Speed–Time Graph when Speed remains Constant

  • If the speed–time graph of a body is a straight line parallel to time axis, then the speed is constant or uniform.



  • Speed is the distance travelled per unit time.

    begin mathsize 12px style Speed equals fraction numerator Distance space travelled over denominator Time space taken end fraction end style
  • Thus, we have Distance travelled = Speed × Time taken.
  • Now, to find the distance travelled at point C, we draw a perpendicular CB.
  • Thus, speed and time at C is

    begin mathsize 12px style Speed space at space straight C space equals space CB space equals space OA space space space space space space space space because CB space equals space OA
Time space at space straight C space equals space OC end style

  • Thus, the distance travelled is
    Distance travelled = OA × OC = Area of rectangle OABC
  • Thus, in a speed–time graph, the area enclosed by the speed–time curve and the time axis gives the distance travelled by the body.
  • Also, we can conclude that the velocity–time graph of an object moving with constant velocity is a straight line parallel to the time axis.
  • The area enclosed by the velocity−time curve and the time axis gives the displacement of the body.
 
Speed–Time Graph when the Speed Changes at a Uniform Rate
 
  • The speed–time graph of a body with uniform changing speed will be a straight line.