Motion

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.

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.
  
  

 

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.
  
  
• If u is the initial velocity and v is the final velocity for a given time interval t, then the average velocity  of the object is given as
      

 

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.
  
  
  
  
• 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
  
  
  
• 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.
  
  
  
  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.
  
  
  
• From the graph, we have
  
  
• But, the slope of the graph = 
• 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.
  
  
• 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
  
  
  
  
• 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.
  
  
  
  
• Acceleration is change in speed per time taken for the change.
  
  
  
• From the graph, we get the change in speed and time as
  
  
  
  
• But the slope of the graph = 
• Thus, the slope of the speed–time graph gives the acceleration of the object.
• Also, distance can be calculated from the speed–time graph as
  
  
  
• The speed–time graph of an object showing upward slope gives uniform acceleration, and similarly, the speed–time graph of an object showing downward slope shows uniform retardation.

 

Speed–time graph when the initial speed of the body is not zero

• The speed–time graph for an object moving with uniformly changing speed and initial speed not equal to zero is shown.
  
  
  
  
  
• The acceleration in this case is
  
  
• Distance can be calculated from the graph as
  
  
  

 

Speed–Time Graph when the Speed Changes at a Non-Uniform Rate

 

• When an object is moving with non-uniform changing speed, then the speed–time graph is a curved line showing non-uniform acceleration.
  
  

 

Acceleration-Time Graph 

 

• In this graph, acceleration is plotted along Y-axis and time along X-axis.
• When body is moving with uniform acceleration, a is constant.
  
  
• Body moving with constant velocity, a = 0
  
   
• Uses of Acceleration–time Graph:
  1. Acceleration at any intermediate time can be determined.
  2. Area under the curve gives the change in magnitude of velocity i.e. (v − u).

 

Equations of Motion

• The three equations of motion can be derived with the help of velocity (speed) – time graph as follows:
• Consider the velocity–time graph of a body with initial velocity u at point A.
  
  
• It has an uniform acceleration from A to B as its velocity changes uniformly in time t. After the time t, its final velocity is v.
• In the graph, u = OA, v = CB, t = OC.
• We draw perpendicular CB from point C and draw AD parallel to OC. Also, BE is the perpendicular from B to OE.

 

First Equation: v = u + at

• Initial velocity of the body is u = OA and final velocity is v = CB.
• From the graph, we have CB = CD + DB
• Therefore,  v = CD + DB
• Now, from the graph, we have DC = OA.
• Therefore,  v = DB + OA.
• Since u = OA, we have
  v = DB + u…… (1)
• Now, we know that the slope of the velocity–time graph gives the acceleration a.
• Thus, we have
  a = Slope of line AB
  
  
   
• But from the graph, AD = OC = t. Thus, acceleration is written as
  
  
• Thus, substituting equation (2) in (1), we get v = u + at.
• This is how the first equation of motion is derived from the graphical method.

 

Second equation: s = ut + ½ at²

• Suppose a body travels a distance s in time t. The distance travelled is given as the area between the graph AB and the time axis OC, i.e. the distance travelled is equal to the area of the figure OABC.
  
  Distance travelled = Area of OABC
                            = Area (rectangle OADC) + Area(Triangle ABD)
• Now, the area of the rectangle is given as
  Area(rectangle OADC) = OA × OC
                                  = u × t = ut
• The area of the triangle is given as
  
  
  
  
  
• Thus, the distance travelled is .

Third equation: v² = u² + 2as

• The distance travelled is equal to the area of the figure OABC. Because OABC is a trapezium,
  
  
  
  
  
• Now, we have OA + BC = u + v  and OC = t . Thus, the distance travelled is
  
  
  
• Eliminate t from the above equation with the help of the first equation of motion as
  v = u + at
  at = v - u
  
  
  
• Substitute equation (2) in equation (1).
  
  
  
  
  
• Thus, the third equation is v² = u² + 2as.

 

Equations of motion under the influence of gravity

• The equation of acceleration due to gravity is . From this equation, we can conclude that g is independent of the mass of the object that is falling. Thus, any object big or small, solid or hollow, should fall with the same acceleration.
• The equations of motion become valid for a falling object as g is constant near the Earth. These equations in terms of g are
  
  
  
  
• Here, g should be positive if it is in the direction of motion, and it should be negative if it moving against the acceleration due to gravity.
• Equations of motion of body dropped from a certain height: -
  u = 0,
  s = h (height from which the object is dropped)
  a = +g
  Thus, 
  1. v = gt
  
  
  3. v² = 2gh
  Thus, time of descent is
  
  
• Equations of motion of body when it is projected upward: -
  s = h (height from which the object is dropped)
  a = – g
  Thus,
   
  For body projected vertically upward, the time of ascent (ta) is time taken to reach the maximum height.
  
  ta = 
  A vertically projected body after reaching its maximum height (h_max) starts falling and behaves as a body dropped from maximum height. The time that it takes to reach the ground from the maximum height, can be given by,
  Thus,
  
  
  
  Thus, time of ascent is equal to time of descent when body is projected vertically up.

 

Time of flight (t_f)

The total time during which a body moving under gravity remains in air above the plane of projection.

 

 

 

Uniform Circular Motion (U.C.M.)

• When an object moves in a circle, its motion is called circular motion.
• Examples
  oMotion of a giant wheel in a funfair.
  
  
  
  oMotion of the wheel of a bicycle.
  
  
  
  
• When an object moves along a circular path, its direction of motion as well as speed changes continuously.
• Thus, the motion is said to be an accelerated motion as the speed (or velocity) keeps changing.
• Now, when the body moves in a circle with uniform speed, its motion is known as uniform circular motion.
• This may suggest that the motion is not accelerated. However, the direction of speed changes at each and every point along the circle. Thus, even though the speed is constant, its direction changes. Hence, the velocity changes and is an accelerated motion.
  
  
  
  
• The force needed to make an object travel in a circular path is known as centripetal force.

 

Examples of Uniform Circular Motion

• Moon revolving around the earth: The moon revolves around the earth in a circular orbit. It completes one revolution of its orbit in a period of about 29.3 days.
• The tip of the seconds’ hand of a clock: The tip of the seconds’ hand of a clock shows uniform circular motion. It completes every circle in a period of 60 seconds.

 

Speed of an Object in Uniform Circular Motion

• When an object completes one round around the circle, it traverses a distance equal to the circumference of the circle.
• Let an object move in a circle of radius r. Thus, the circumference is equal to 2πr.
• Now, the speed of the object is given as
  
  
  
• Thus, the speed of the object performing uniform circular motion is
  
  
  .