Question

True or False If the velocity of a particle moving along the \(x\) -axis is always positive, then the displacement is equal to the total distance traveled. Justify your answer.

Short answer

The statement is true. If the velocity of a particle moving along the x-axis is always positive, then the displacement equals the total distance traveled.

Step-by-step solution

Understanding the Definitions

Displacement is a vector that points from an object's initial position to its final position and has a magnitude equal to the shortest distance between two points. The total distance travelled is the length of the actual path travelled by the object. For any object moving along the x-axis, if the object moves in a back and forth manner but its velocity is always positive, then it means the object is always moving in the forward direction on x-axis.

Evaluating the Statement

Since the velocity is always positive, the object is always moving forward. Therefore, the distance it covers during the journey is in the same direction. This means the total distance equals the displacement.

Draw the Conclusion

The displacement is a shortest path from the start point to the end point. Since in this case, it's same as the total path traveled because the velocity is always positive and thus the object is always moving forward, it can be concluded that the statement is true. If the velocity is always positive, then the displacement is equal to the total distance traveled.

Key concepts

Velocity and Displacement

Imagine you're sending a paper airplane on a journey across a room. If we want to map its flight, we'll use two concepts: velocity and displacement. Velocity is the speed of an object in a specified direction. For the paper airplane, it's the speed at which it's flying forward. But what about displacement? This is where things get interesting in calculus!

Displacement measures how 'out of place' an object is after its journey, relative to where it started. It doesn't care about the path, only the start and finish. In our paper airplane example, even if the plane goes in circles before landing, the displacement is a straight line from the launch point to the landing point. This is why we often describe displacement using vectors in calculus – they give direction as well as magnitude.

If you throw your paper airplane and it always goes forward, never backward, its velocity is always positive in the direction of flight. Consequently, the airplane's displacement will directly correspond to the total distance it has flown, because there's no 'backtracking' – the forward flight path is the displacement.

Displacement vs Total Distance

Sometimes, understanding the difference between displacement and total distance can be as confusing as navigating through a maze. But let's simplify it. Picture yourself walking your dog through the park. You take several turns, maybe circle around a few trees, and double back to check out a squirrel your dog spotted. When you finally stop, the total distance you walked is like the dotted line you have drawn all over the park – it's the complete path you followed.

But your displacement? It's far more straightforward – literally. It's the shortest straight-line distance from your starting point to your stopping point. What if you've only walked forward, never backward? Well, then your squirrely journey has the same length as your displacement — they're identical.

In calculus terms, if we represent your walk as a graph on the x-axis with time, the total area under the curve will be the total distance. However, if you always walk forward, that area under the curve is also your displacement.

Positive Velocity

Positive velocity is like always knowing one direction: forward. It means our object – it could be a car, a runner, or even a river – is moving towards a particular destination without turning back. Picture a runner on a straight track, maintaining a steady pace forward; that's positive velocity in action. It doesn't just tell us the object is moving – it shows the direction is consistently the same.

In the realm of calculus, if we plot the position of our runner over time, the slope of the position-time graph gives us the velocity. A positive slope, therefore, signifies a positive velocity, indicating that as each second goes by, our runner is covering more ground in the forward direction. And here's an interesting nugget: with a constant positive velocity, not only is our runner always moving forward, but the total distance covered and the displacement will be the same – because there's no reverse gear in their journey.