Speed, Velocity and Acceleration (Kinematics - 1)

Rest and Motion:

We can say if a body does not change it's position with time, it is at rest. And if a body changes it's position with time, it is said to be moving. But, how do we know that a particle is in motion or not ?

To find the position of the particle we need a frame of reference. Let's fix the frame of reference as the three mutually perpendicular axes X, Y and Z. Assume that particle has co-ordinates (x, y, z) at any instant. After a while, if all co-ordinates of the particle remains same with time that means particle is at rest with respect to that frame of reference. And if one or more co-ordinates change with time, it is called as moving particle.

There is not any restriction of a choice of a frame of reference. We can choose a frame of reference according to our convenience. For simplicity assume yourself at rest at the frame of reference. For example a bus is moving with a speed. First, I take bus as the reference that means I should have put myself in the bus. I see, every person in the bus is at rest with respect to me while bus is moving. Second, I consider road as a frame of reference that puts me fix at the road. Now, every person in the bus seems moving with the bus. Now remember your one visit in car or bus or train. You must did observe that every tree, tower, place were moving with respect to you. You know that they are static. They can't move. We did see them as moving because we are in moving bus or car or whatever and that is reference to us.  

Distance and Displacement: 

Suppose a particle starts from position A and move along the green path to reach at position B. Here length of the green path is known as distance traveled by the particle. If we connect the initial position A with the final position B by a straight line, we get the

displacement of the particle. The magnitude of the displacement is the length of straight line (red line in diagram), which is shortest distance between position A and position B. The direction of the displacement is from initial position A to final position B.

Displacement has both magnitude as well as direction. That means displacement is a vector quantity. Whereas, distance has only magnitude that's why it is known as scalar quantity. S.I. unit of distance and displacement is meter(m) and dimension is [L].

Example 1: A man walk from position A to position E along the given path. 

Find the displacement and distance traveled by the man ?

Solution: 

Total distance traveled by the man, d = AB + BC + CD + DE

                                                         d = 5 m + 5 m + 4 m + 6 m

                                                         d = 20 m

In this case displacement would be straight length between initial position A and final position E.

           In the right angle triangle BCD 

                                  BC = 5 m , CD = 4 m

                                      so, BD = √5²-4² = √9 = 3 m

           Similarly, ADE is also a right angle triangle, in this

                                  AD = AB + BD = 5 + 3 = 8 m

                                         and DE = 6 m 

                                  then AE = √8²+6²
                                              AE = √100 = 10 m
So, displacement of the man is 10 m, whereas distance is 20 m.

Note:

• Displacement of the particle can be zero but distance has always non zero value.
• Displacement can be negative, distance has only positive values. 

       Speed, Velocity and Acceleration:

Speed:

Distance traveled by the particle in a given time interval is known as the average speed. If a particle traveled s distance in a time interval t, then average speed -

                                          vavg = s/t

The average speed gives the total effect in the given time interval.

Consider that ∆s is the distance traveled by the particle in the time interval t to t+∆t. The average speed would be -
                                      vavg = ∆s/∆t

If we decrease the time interval ∆t, the distance ∆s will also decrease. As ∆t approaches 0, the distance ∆s also approaches 0. But the ratio ∆s/∆t has a finite limit, which is called as instantaneous speed. It can be called as speed.

         here, s is the distance traveled in time t.  

S.I. unit of speed is meter/second(m/s) and dimension is [LT¯1 ].

Note:

• The average speed is defined for a time interval and the instantaneous speed is defined at a particular instant. 

Example 2: A car is moving with a speed s = (5 - 2t)t . Find the average speed of the particle in the time period t = 3s to t = 5s, and instantaneous speed at t = 2s.
Solution:

Velocity:

Average velocity of the particle is defined as average rate of change of displacement in the given time interval while the instantaneous velocity is the specific rate of change of displacement at a particular instant.

A particle moves from position A(r₁)to position B(r₂) in a time interval t₁ to t₂. 

The displacement vector would be AB(r₂₁). So average velocity of the particle is -

Consider that ∆r is the displacement of the particle in the time interval t to t+∆t. So, average velocity can be written as
                                  vavg = ∆r/∆t
If ∆t tends to zero then displacement vector ∆r is also tends to zero. So limiting value of the average velocity is known as instantaneous velocity at a particular time t. Instantaneous velocity is called velocity.

S.I. unit of velocity is meter/second(m/s) and dimension is [LT¯1 ]

For a very small time interval displacement ∆r equals to the distance ∆s. Since displacement is a vector quantity so magnitude of displacement can be written as distance.
                                         |∆r| =  ∆s
Similarly, magnitude of the velocity is speed.

Acceleration:

Rate of change of velocity is Acceleration. If any particle does not change its velocity with the time, it is said to be moving with a uniform velocity. Suppose the velocity of a particle at time t1 is v1 and at time t2 it is v2. Average acceleration of the particle is define as the ratio of the change in velocity and change in time.

Instantaneous acceleration is the specific rate of change of velocity at a instant. Again same, if ∆v is the change in velocity in the time t to t+∆t, then for ∆t tends to zero change in velocity(∆v) also tends to zero. And in this case limiting value of the average acceleration is instantaneous acceleration. Instantaneous velocity is called acceleration.

S.I. unit of acceleration is meter/second²(m/s²) and dimension is [LT¯² ].