Question

The displacement $ x $ versus time graph is shown below.
 
The displacement $ x $ is plotted against time $ t $. Choose the correct answer from the options given below:

• A. The average velocity during 0 to 3 s is 10 m/s
• B. The average velocity during 3 to 5 s is 0 m/s
• C. The instantaneous velocity at \( t = 2 \) s is 5 m/s
• D. The average velocity during 5 to 7 s is the same as instantaneous velocity at \( t = 6.5 \) s

Hint:

The average velocity is the total displacement divided by the total time, while instantaneous velocity is the slope of the displacement-time graph at a given point.

Answer 1

The Correct Option is D

To address this issue, we must examine the provided displacement-time graph and understand the definitions of average and instantaneous velocity.

Definitions

• Average Velocity: This is calculated as total displacement divided by the total time elapsed. Mathematically, it is expressed as \(v_{\text{avg}} = \frac{\Delta x}{\Delta t}\).
• Instantaneous Velocity: This represents the velocity of an object at a precise moment in time. It is determined by calculating the slope of the tangent line to the graph at that specific point.

Analysis of Options

1. The average velocity during 0 to 3 s is 10 m/s:
   During this interval, the object transitions from point A (at 0 s, position \(x = -5 \, \text{m}\)) to point C (at 3 s, position \(x = 5 \, \text{m}\)).
   The total displacement is \(5 \, \text{m} - (-5 \, \text{m}) = 10 \, \text{m}\).
   The average velocity is thus \( \frac{10 \, \text{m}}{3 \, \text{s}} = \frac{10}{3} \, \text{m/s} \), which is not equal to 10 m/s.
   This statement is incorrect.
2. The average velocity during 3 to 5 s is 0 m/s:
   Between 3 s and 5 s, the object moves from point C (position 5 m) to point D (position 5 m). Consequently, there is no change in displacement.
   Total displacement = \(5 \, \text{m} - 5 \, \text{m} = 0 \, \text{m}\).
   Average velocity = \( \frac{0}{2 \, \text{s}} = 0 \, \text{m/s} \).
   This statement is correct. However, we will evaluate the remaining options.
3. The instantaneous velocity at \( t = 2 \) s is 5 m/s:
   At \( t = 2 \, \text{s} \), the graph displays a linear segment (A to C) with a calculated slope of \(5 \, \text{m/s}\).
   This statement is consistent with the graph, but it may not be the sole correct assertion.
4. The average velocity during 5 to 7 s is the same as instantaneous velocity at \( t = 6.5 \) s:
   From 5 s to 7 s, the object travels from point E (position 10 m at 5 s) to point G (position 0 m at 7 s).
   Total displacement = \(0 \, \text{m} - 10 \, \text{m} = -10 \, \text{m}\).
   Average velocity = \( \frac{-10 \, \text{m}}{2 \, \text{s}} = -5 \, \text{m/s} \).
   At \( t = 6.5 \) s, the slope of the graph segment EF is -5 m/s.
   This statement is correct and aligns with the indicated correct answer.

Conclusion

The correct statement is: The average velocity during 5 to 7 s is the same as instantaneous velocity at \( t = 6.5 \) s.