Sunday, January 15, 2012

Speed, velocity and acceleration from scratch




  Average speed
 
    Unless he or she is traveling on a superhighway, a motorist cannot usually maintain a constant speed for any length of time. In the ordinary way, traffic conditions frequently cause him or her to change speed or stop.
   
     When deciding the time to allow for a particular journey, the motorist must therefore have some idea of the average speed at which he or she swill be able to travel.
    
    Speed is defined as the rate of change of distance moved with time.
   
Thus, if a journey distance is 160 km, takes four hours,

          average speed = distance / time = 160 / 4 = 40 km / h
         
During such a journey, however, the actual speed of the car at any instant will vary considerably from this figure.






Actual speed

  Normally, a driver notes his actual speed at any moment by glancing at the speedometer, but speedometer readings are not particularly reliable.

In order to obtain an accurate value for the speed of a vehicle at any instant it would be necessary to measure the distance moved in a very short interval of time.

This is best done by a roadside observer using special apparatus to time the car over a measured distance. So long as the time interval is short, there is less likelihood that the speed will vary over the measured distance. Otherwise, of course, the value obtained will be an average speed.

 
  Distance and displacement

Let's first explain the difference between scalar and vector quantities.



     A scalar quantity is one which has only magnitude ( or size). A vector quantity has both magnitude and direction. Thus, when we say that a library has 1000 books or a tank contains 100 liters of petrol, we are dealing with scalar quantities.

On the other hand, it would be of little use to talk about a force of 10 newtons unless we also state the direction in which it acts, e.g., 10 N vertically downwards. Force is there a vector.

    Examples of scalar quantities are length, area, time, distance, mass, speed, pressure, temperature,  work, power, and density.

    Examples of vector quantities are displacement, velocity, acceleration,  momentum, thrust, and force.

    Now, if we say that a car travels a distance of 100 m, the expression " 100 m " is a scalar quantity. But if the car happens to be moving along a straight line and we mention the direction of travel, e.g., 100 m due east, we are now dealing with a vector quantity, and this is called the displacement of the car.
   
    Displacement is defined as distance moved in a specified direction.


Velocity

  In ordinary conversation the word " velocity" is often used in place of speed. In science, however, it is important to distinguish between these two terms.
 
  Velocity is defined as the rate of change of distance moved with time in a specified direction ( or, rate of change of displacement).
 
  Velocity is therefore a vector, whereas speed is a scalar quantity. For example, if a car were traveling at a steady speed of 60 km / h along a perfectly straight road, it would be correct to say that it had a velocity of  60 km / h east of north, or whatever the direction of the road might be.

   On the other hand, if  the car were traveling round a bend road with constant speed, its direction of motion would be continuously changing. Hence its velocity would also be continuously changing, although the speed remains constant.
 
  Uniform velocity
 
    A body is said to move with uniform velocity if its rate of distance moved with time in a specified direction is constant.
   
    Acceleration
   
      When the velocity of a body is changing, the body is said to be accelerating.

      Acceleration is defined as the rate of change of velocity with time.


    Acceleration is regarded as positive if the velocity is increasing, and negative if the velocity is decreasing.

Ordinarily, however, most people restrict the use of the word acceleration to cases of  increasing velocity only, while a decreasing velocity or slowing down is usually called a deceleration or retardation.
    
     Worked example
      A motorcar is uniformly retarded and brought to rest from a speed of 108 km / h in 15 s. Find its acceleration
    
     Solution
    
     108 km / h = 108 × 1000 / 60 × 60 = 30  m /s
     therefore
                      initial velocity = 30  m /s
                      final velocity   = 0  m /s
                  change in velocity = final velocity   -  initial velocity
                                                     = 0 - 30 = - 30 m /s
                  acceleration          = change in velocity / time = - 30 m /s  / 15 s = - 2 m/s²
The minus sign here simply means that the car is accelerating in the opposite direction to its initial velocity.
      Coming up : Equations of uniformly accelerated motion with more worked examples. Good luck.