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Motion in One Dimension Study Guide

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Updated on Sep 26, 2011

Introduction

We start the article by defining the notions of displacement, velocity, speed, and acceleration, Then we look into graphical representation of motion as a way of describing the motion of an object or particle, We consider a special case of unidimensional motion where the acceleration is constant and show an application to the free fall of objects in the gravity field of Earth,

Displacement, Velocity, and Acceleration

Motion involves an object changing its position during some finite time interval with respect to a given reference frame. For most of the time, the object's position may be associated with just one set of coordinates at a time—that would be just one coordinate in the case of a one-dimensional motion. Also, motion is a relative notion: You may be sitting still in a car, so you do not move, but the car may be speeding on the highway, and therefore, you are in fast motion with respect to the road.

We say that displacement is the change in position of a particle or object with respect to a given reference frame. Because the position is a vector, the displacement is also a vector: It points from the initial to the final position and has a magnitude equal to the distance between the two positions. We can always associate both a direction and a size with a displacement.

For a particle moving along a straight line in one dimension, the displacement can be expressed in terms of the initial and final coordinates as:

Δx = xfxi

If xf is greater than xi, then we have a positive displacement; otherwise, the displacement is negative.

For any object in motion for a finite duration of time, we can define an average speed as the ratio of the distance traveled over the time elapsed:

This formula assumes our moving object starts at the time zero at some point P and then moves a distance d during a time interval t.

Example 1

You can go from Lansing to Detroit, Michigan, in, say, 2 hours, but out of these 2 hours, you spend 1 hour going 60 miles from Lansing to Novi and the next hour going the rest of 40 miles from Novi to Detroit (mostly stuck in traffic). Find the average speed from Lansing to Novi, the average speed from Novi to Detroit, and the overall average speed from Lansing to Detroit.

Solution 1

Since all quantities are given in miles and hours, we will keep these units instead of using SI units.

Your average speed from Lansing to Novi is 60 miles pee hour. From Novi to Detroit, as you are stuck in traffic, your average speed is only 40 miles per hour. The overall average speed from Lansing to Detroit is:

We note that for calculating the average speed, only the total distance traveled and the total time are important. The example also shows that during a long time interval, we may at times move faster and at other times slower; the overall average speed gives us an idea of the rate at which we move on average during the whole trip.

Now let's look for a moment at what happens when the time interval t is made almost zero. In this case, the distance traveled d also becomes very small (for normal-range finite speed values). Therefore, the start point and the end point are very close to each other. Our average speed is calculated over a very small space interval around the point P. In this case, our average velocity is a very close approximation to the instantaneous speed at point P.

Back to our example, a good practical approximation of the car's instantaneous speed at every moment of time during the trip from Lansing to Detroit is the value we read on the car's speedometer.

Let's note that speed is always a positive quantity. The 51 unit for measuring speed follows from its definition: meter per second. Other convenient units may be miles per hour or kilometers per hour.

As far as the direction of the movement is concerned, we need to extend the speed to the concept of velocity. We can reach this by replacing the "distance" in the definition of the average speed with the "displacement" vector. The instantaneous velocity is a vector obtained by taking the ratio of the displacement vector d to the time t elapsed for that displacement to occur, when the time interval is very small:

Therefore, the instantaneous velocity indicates how fast the car moves and the direction of motion at each instant of time.

For the case of motion in one dimension along a straight line, the distance and the magnitude of the displacement are the same, and therefore, the speed is the magnitude of the velocity. This statement remains true in the general case of an arbitrary motion in two or three dimensions. The proof is a trivial exercise of infinitesimal calculus. Is it really a surprise that Sir Isaac Newton is, at the same time, one of the inventors of the infinitesimal calculus and the famous father of the laws of mechanics?

Instantaneous Velocity

Instantaneous velocity indicates how fast the ear moves and the direction of motion at each instant of time.

As the velocity shows the change in an object's position, the acceleration shows how the velocity of the object changes. As the velocity is a vector, it can change either its magnitude or its direction. In either case, the object will feel an acceleration a given by:

Therefore, in the International System, or SI, unit for acceleration is m/s2.

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