Relative Motion in One Dimension

Earth always seems to be stationary to human beings, but in reality, Earth is constantly revolving around the Sun, and it is a Universal Truth. Then, why do Humans not feel the Earth moving? The answer is simple- Relative Motion.

It means that the motion is always relative in Practical life. An object that is moving with a speed of v to a certain person means that the speed is measured while taking the person as a frame of reference. Similarly, when we are moving in a vehicle at a certain speed, the ground and the trees seem to be moving backward, but in reality, they are stationary and cannot move. These are real-life examples and proof of the existence of Relative motion.

Relative Motion

The concept of reference frames was first introduced to discuss relative motion in one or more dimensions. When we say an object has a certain velocity, that velocity is always measured with respect to a specific frame of reference. In everyday situations, we usually take the reference frame to be the ground or the Earth.

For example, if you are traveling on a train moving at 100 km/h, a passenger sitting on the train would see your speed as zero since you're both moving together. However, if someone is observing you from outside the train, standing on the ground, they would see you moving at 100 km/h because you are on the train, which is moving at that speed.

The motion observed by an observer depends on their location (or frame of reference). This type of motion is known as relative motion.

Also Read, Frame of reference

Relative Velocity

Suppose there are two objects P and Q moving at their respective speeds of v_p and v_q, now according to their positions, their relative velocities will be different. Suppose their initial displacement is X₀₁ and X₀₂ and their final displacement is X₁ and X₂ in time t when they are moving in the same direction, 

Then, X₁= X₀₁ + v_pt  ⇢ (1)

X₂= X₀₂+ v_qt  ⇢ (2)

Subtract (1) from (2),

X₂ - X₁ = X₀₁ - X₀₂ + (v_qt- v_pt)

X₂ - X₁ = (X₀₂ - X₀₁) + (v_q- v_p)t

The final Relative displacement is given as, X = X₂ - X₁ (Displacement of object Q w.r.t. P)

The Initial Relative displacement is given as, X₀ = X₀₂ - X₀₁ (displacement of object Q w.r.t. P)

The relative velocity is given as, v = v_q- v_p (Velocity of object Q w.r.t. object P)

Graphs of Relative Motion

The relative velocity may Vary in value and can become positive, negative, or even zero based on the absolute values of the objects. If two velocities v₁ and v₂ are in the same direction, the relative velocity of object 2 w.r.t 1 is (v₂- v₁). When these objects are moving in opposite directions, the relative velocity becomes (v₂ + v₁).

Let's look at different cases based on the value of velocities (v_p and v_q) of a different object when the objects are moving in the same direction,

Case 1: The objects have the same velocities (v_p = v_q)

When the two objects have same velocity, and they are moving in the same direction, the distance between both of them will always remain constant (X₂ - X₁ = X)

Case 2: When the second object has more velocity than the first object (v_q > v_p)

When the second object (object Q) has more velocity than the first object (P), since Q is ahead of P, the distance between them will always be positive as P will not be able to cross Q.

Case 3: When the first object (P) has more velocity than the second object (v_p > v_q)

When the second object has lesser velocity, object P will be easily able to cross object Q and hence, the difference between the distance of the second object w.r.t first object will be negative.

Note:

• When the objects are moving in the same direction, the magnitude of the relative velocity between them is the sum of the velocities of the objects.

• When the objects are moving in the opposite direction, the magnitude of the relative velocity between them is equal to the difference between the magnitude of their velocities.

Solved Examples of Relative Motion in One Dimension

Example 1: A particle is moving with some constant velocity of v m/sec if it moves with the time t and covers a distance of x(t), and it is given that, x(1) = -2, x(7) = 6. Find,

1. The displacement covered by the particle from t = 1 second to 7 seconds.
2. The value of the constant velocity (v m/sec).

Solution:

1. Distance at t = 1 second, x(1) = -2

Distance at t = 7 seconds, x(7) = 6

Total displacement = 6- (-2) = 6+2 = 8 units

2. Constant velocity = Displacement/ Time = 8/(7-1) = 8/6 = 4/3 m/sec. 

Example 2: Rahul is riding his bicycle at a speed of 6m/sec and Raman is jogging at a speed of 2m/sec in the same direction. Prove that the relative speed of Raman w.r.t Rahul will be negative.

Solution:

Raman's speed w.r.t. Rahul = (Speed of Raman)- (Speed of Rahul)

= 2 m/sec - 6 m/sec

= -4 m/sec

Hence, The Relative speed of Raman w.r.t. Rahul is negative.

Example 3: In the Previous Question, Hypothetically Imagine that both Rahul and Raman are on a very large platform that is moving with a speed of 10,000m/sec. Will this newly added factor have any change in the relative speed between them provided the absolute speeds are the same as before?

Solution:

There will be No change in the relative speed between them because of the introduction of large platform moving with a speed of 10,000m/sec, since the platform is under both Rahul and Raman. In other sense, Earth too acts as a very big platform which constantly moves around the Sun but the formula for Relative velocity still remains the same.

Related Reads

• Frame of Reference
• Relative Motion