Problem 25 - Master Resource Book in JEE Main Physics

Question

A body is moving along a straight line path with constant velocity. At an instant of time the distance of time the distance travelled by it is \(s\) and its displacement is \(D\), then (a) \(D5\) (c) \(D=s\) (d) \(D \leq s\)

Step-by-Step Solution

Answer

The correct answer is (c) \(D = s\).

Step 1: Understanding Key Terms

To solve this problem, we need to understand the difference between distance and displacement. **Distance** (\(s\)) is the total length of the path traveled, while **displacement** (\(D\)) is the straight-line measurement from the initial to the final position, with direction included. In the case of straight-line motion with a constant velocity, these two measurements are particularly interesting.

Step 2: Analyzing Straight Line Motion with Constant Velocity

Since the body is moving with a constant velocity along a straight line, it means there is a uniform motion. In such a scenario, the distance traveled and the displacement become important. Because the path is straight and there's no change in direction, the magnitude of the distance traveled should equal the magnitude of the displacement, \(s = D\).

Step 3: Checking All Options

Now, let us check each of the given options against what we analyzed:- **Option (a):** \(D < s\) is not valid for straight-line constant velocity motion without changes in direction.- **Option (b):** \(D > s\) makes no logical sense as discussed.- **Option (c):** \(D = s\) aligns perfectly with our analysis where distance and displacement are equal in magnitude.- **Option (d):** \(D \leq s\) maps scenarios with possible curves or directional changes but simplifies to \(D = s\) here due to the constant velocity on a straight line.

Step 4: Conclusion

For motion along a straight line with constant velocity, the correct relationship between distance and displacement is \(D = s\) because there are no turns or direction changes to affect the displacement. The body moves continuously in one direction, making displacement equal to the distance traveled.

Key Concepts

Constant Velocity

When we talk about constant velocity, we are describing motion where an object moves at a steady speed in a straight line. This means there is no acceleration or deceleration involved, and the speed remains the same over time.

It's important to remember:

• Velocity includes both the speed and direction of motion.
• A constant velocity implies uniform motion without any change in either velocity or direction.
• If velocity changes, even slightly, it is no longer considered constant velocity.

In the context of straight-line motion, constant velocity ensures that the relationship between distance and displacement remains straightforward. Because velocity is constant and the path is direct, both distance and displacement reflect the simplest measurement possible.

Distance vs Displacement

Many students might find it challenging to distinguish between distance and displacement, but it's crucial to understand the difference.

- **Distance** is the total length traveled by an object, regardless of its starting or ending point. It is scalar, which means it has only magnitude—no direction. - **Displacement** refers to the change in position from the starting point to the ending point in a straight line. It is a vector quantity, possessing both magnitude and direction.

In straight-line motion with constant velocity, these two can be considered equal since there are no changes in direction. This balance makes problems easier to solve because we can confidently say that the length of the path traveled (distance) perfectly matches the direct measurement from start to end (displacement).

Uniform Motion

Uniform motion is an essential concept in understanding straight line motion, especially when discussing objects moving at constant velocity.

- **Uniform motion** happens when an object travels equal distances in equal time intervals. In simpler terms, the speed doesn't vary—it's consistent throughout the motion.

In real life, certain conditions like friction or external forces can cause deviations from uniform motion. However, in theoretical physics or ideal conditions, objects in uniform motion retain their constant speed and follow a direct path.

Understanding uniform motion allows for better comprehension of problems involving constant velocity, since it supports the idea that both distance and displacement remain the same when direction doesn't change.