Measuring the Rate of Change of Motion

Motion is common to everything in the universe. We walk, run and ride a bicycle. Even when we are sleeping, air moves into and out of our lungs and blood flows in arteries and veins. We see leaves falling from trees and water flowing down a dam. Motion is change in position of an object with time. How does the position change with time ? In this chapter, we shall learn how to describe rate of change in motion.

Average Velocity

When an object is in motion, its position changes with time. But how fast is the position changing with time and in what direction? To describe this, we define the quantity average velocity

• Average Velocity = Change in position/time interval
• Vavg = (xf-xi)/(tf – ti)
  
• Where xf and xi are the positions of the object at time tf and ti , respectively.The SI unit for velocity is m/s or m s–1, although km h–1 is used in many everyday applications.
• 1km/hr = 5/18 m/sec.

Average Speed

The magnitude of displacement may be different from the actual path length. To describe the rate of motion over the actual path, we introduce another quantity called average speed. Average speed is defined as the total path length travelled divided by the total time interval during which the motion has taken place :

• Average speed = Total path length/Total time interval
• SI unit = m s–1
• It does not tell us in what direction an object is moving.
• It is always positive (in contrast to the average velocity which can be positive or negative).
• If the motion of an object is along a straight line and in the same direction, the magnitude of displacement is equal to the total path length. In that case, the magnitude of average velocity is equal to the average speed.
• If the motion of an object is not along a straight line and it’s motion may be same or different, the magnitude of displacement is not equal to the total path length. In that case, the magnitude of average velocity is not equal to the average speed.

Velocity (Rate of change of position)

The average velocity tells us how fast an object has been moving over a given time interval but does not tell us how fast it moves at different instants of time during that interval. For this, we define instantaneous velocity or simply velocity “v” at an instant “t”. The velocity at an instant is defined as the limit of the average velocity as the time interval ∆t becomes infinitesimally small. In other words,

• vinst = dx/dt
• Where (dx/dt) is the rate of change of position with respect to time, at that instant.
• The SI unit for velocity is m/s or m s–1.

Instantaneous Speed or Speed

• Instantaneous speed or simply speed is the magnitude of velocity.
• Example: A velocity of + 24.0 m s–1 and a velocity of – 24.0 m s–1; Both have an associated speed of 24.0 m s–1.
• It should be noted that though average speed over a finite interval of time is greater or equal to the magnitude of the average velocity.
• Instantaneous speed at an instant is equal to the magnitude of the instantaneous velocity at that instant.
• SI unit = m/sec
• It does not tell us in what direction an object is moving.