Chapter 3.9 - Relative velocity - Special case

In the previous section we saw details about relative velocity in rectilinear motion. In this section we will see a special case.

Consider fig.3.55(a) below:

Fig.3.55

1. A 'platform on wheels' of very long length is moving towards the positive side of the x axis.
• The reading shown by the speedometer of the moving platform is V_P ms^-1
• So the platform is moving with a velocity of V_P ms^-1. 
• It is moving on a railway track
2. A bridge runs directly above and parallel to the railway track. On that bridge, a car is travelling towards the positive side of the x axis
• The reading shown by the speedometer of the car is V_C ms^-1
• So the car is moving with a velocity of V_C ms^-1
3. A point M is marked on the platform
• When viewed from M, the car will appear to be moving with a 'particular velocity'
• This  'particular velocity' is the velocity of the car w.r.t the platform. It is given by: V_CP = (V_C - V_P)
4. Since both the car and the platform travels towards the positive side of the x axis, both are positive values
• So, if V_C is greater than V_P, we get a positive value for V_CP
• That means, when viewed from M, the car appears to be moving towards the positive side of the x axis
5. If V_C is less than V_P, we get a negative value for V_CP
• That means, when viewed from M, the car appears to be moving towards the negative side of the x axis
• That means, when viewed from M, the car appears to be moving further and further backwards
6. Consider the point N marked on the ground
• A stationary observer at N will see the following:
    ♦ The platform moving towards the positive side of the x axis with a velocity of V_P
    ♦ The car moving towards the positive side of the x axis with a velocity of V_C.
• The car will surely pass the obsever, whatever be the velocity of the car

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Now consider fig(b):
1. As before, the platform is moving towards the positive side of the x axis.
• The reading shown by the speedometer of the moving platform is V_P ms^-1
• So the platform is moving with a velocity of V_P ms^-1
2. The car is now moving on the platform
• It is travelling towards the positive side of the x axis
• The reading shown by the speedometer of the car is V_C ms^-1
• So the car is moving with a velocity of V_C ms^-1
3. A point M is marked on the platform
• When viewed from M, the car will appear to be moving with a 'particular velocity'
This 'particular velocity' is nothing but the velocity VC shown by the speedometer of the car
So this time we do not have to make a subtraction
4. Consider the point N marked on the ground
• A stationary observer at N will see the following:
    ♦ The platform moving towards the positive side of the x axis with a velocity of V_P
    ♦ The car moving towards the positive side of the x axis with a velocity of (V_C + V_P)
• The car will surely pass the obsever, whatever be the velocity of the car

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Consider fig.3.56(a) below:

Fig.3.56

1. A 'platform on wheels' of very long length is moving towards the positive side of the x axis.
• The reading shown by the speedometer of the moving platform is V_P ms^-1
• So the platform is moving with a velocity of V_P ms^-1. 
• It is moving on a railway track
2. A bridge runs directly above and parallel to the railway track. On that bridge, a car is travelling towards the negative side of the x axis
• The reading shown by the speedometer of the car is V_C ms^-1
• So the car is moving with a velocity of V_C ms^-1
3. A point M is marked on the platform
• When viewed from M, the car will appear to be moving with a 'particular velocity'
• This  'particular velocity' is the velocity of the car w.r.t the platform. It is given by: 
V_CP = (-V_C -V_P) = -(V_C + V_P)
• V_C is given a negative sign because, the car is moving towards the negative side of the x axis  
4. Since V_CP is negtive, we can write:
When viewed from M, the car appears to be moving towards the negative side of the x axis
5. Consider the point N marked on the ground
• A stationary observer at N will see the following:
    ♦ The platform moving towards the positive side of the x axis with a velocity of V_P
    ♦ The car moving towards the negative side of the x axis with a velocity of V_C.
• The car will surely pass the obsever, whatever be the velocity of the car

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Now consider fig(b):
1. As before, the platform is moving towards the positive side of the x axis.
• The reading shown by the speedometer of the moving platform is V_P ms^-1
• So the platform is moving with a velocity of V_P ms^-1
2. The car is now moving on the platform
• It is travelling towards the negative side of the x axis
• The reading shown by the speedometer of the car is V_C ms^-1
• So the car is moving with a velocity of V_C ms^-1
3. A point M is marked on the platform
• When viewed from M, the car will appear to be moving with a 'particular velocity'
This 'particular velocity' is nothing but the velocity VC shown by the speedometer of the car
So this time we do not have to make a subtraction
4. Consider the point N marked on the ground
• A stationary observer at N will see the following:
    ♦ The platform moving towards the positive side of the x axis with a velocity of V_P
    ♦ The car moving towards the negative side of the x axis with a velocity of (V_C - V_P)
• The car will move towards the right and away from the observer, if V_C is less than V_P.
    ♦ The car will never be able to pass the observer in this case
• The car will appear to be stationary, if V_C is equal to V_P

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Now we will see a solved example:
Solved example 3.10
A jet airplane travelling at the speed of 500 $km~h^{–1}$ ejects its products of combustion at the speed of 1500 $km~h^{–1}$ relative to the jet plane. What is the speed of the latter with respect to an observer on the ground ?
Solution:
1. Let us assume that, the jet travels towards the positive side of the x axis
2. Let $V_J$ be the 'actual velocity' of the jet.
• That is., $V_J$ is the velocity of the jet respect to the ground 
• Then we can write: $V_J$ = +500 $km~h^{–1}$ 
3. The products of combustion (smoke) will be moving towards the negative side of the x axis
4. Let $V_S$ be the 'actual velocity' of the smoke. 
• Consider the following two velocities:
(i) 'velocity with which the smoke appears to move', when viewed from the jet 
(ii) 'velocity with which the smoke of combustion appears to move', when viewed from the ground
• Surely, (i) is the 'actual velocity' of the smoke
• So we can write: $V_S$ = -1500 $km~h^{–1}$
5. Each particle of the gas emerges from the jet. But the jet is moving towards the right
• So the observer on the ground sees only a net velocity. 
• We can write: Velocity of smoke with respect to ground = -1500 + 500 = -1000 $km~h^{–1}$
• The negative sign indicates that, the smoke appears ot move with a velocity of 1000 $km~h^{–1}$ towards the negative side of the x axis

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In the next chapter, we will see motion in a plane.

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