MOTION IN ONE-DIMENSION

MOTION IN ONE-DIMENSION
Lecturer : Ibu Vina Serevina

Chapter 2

Motion in One-Dimension

In physics we can categorize motion into threetypes: translational, rotational, and vibrational. A car moving down a highway is an example of translational motion, the Earth’s spin on its axis is an example of rotational motion, and the back-and-forth movement of a pendulum is an example of vibrational motion.

Position, Velocity, and Speed

           

A particle’s position is the location of the particle with respect to a chosen reference point that we can consider to be the origin of a coordinate system. The displacement of a particle is defined as its change in position in some time interval. Therefore, we write the displacement, or change in position, of the particle as

 ∆x ≡ xf – xi

It is very important to recognize the difference between displacement and distance traveled. Distance is the length of a path followed by a particle. Displacement is an example of a vector quantity. Many other physical quantities, including position, velocity, and acceleration, also are vectors. In general, a vector quantity requires the specification of both direction and magnitude. By contrast, a scalar quantity has a numerical value and no direction. In this chapter, we use positive (+) and negative (-) signs to indicate vector direction. We can do this because the chapter deals with one-dimensional motion only; this means that any object we

study can be moving only along a straight line.

The average velocity v–x of a particle is defined as the particle’s displacement ∆x divided by the time interval ∆t during which that displacement occurs.The average speed of a particle, a scalar quantity, is defined as the total distance traveled divided by the total time interval required to travel that distance. However, unlike average velocity, average speed has no direction and hence carries no algebraic sign.

Average Speed and Average Velocity

The magnitude of the average velocity is not the average speed. For example, consider the marathon runner discussed here. The magnitude of the average velocity is zero, but the average speed is clearly not zero.

If the particle moves along a line without changing direction, the displacement and distance traveled over any time interval will be the same. As a result, the magnitude of the average velocity and the average speed will be the same. If the particle reverses direction, however, the displacement  will be less than the distance traveled. In turn, the magnitude of the average velocity will be smaller than the average speed.

Instantaneous Velocity and Speed

The instantaneous velocity vx equals the limiting value of the ratio ∆x/∆t as ∆t approaches zero. The instantaneous velocity can be positive, negative, or zero.  The instantaneous speed of a particle is defined as the magnitude of its instantaneous velocity. As with average speed, instantaneous speed has no direction

associated with it and hence carries no algebraic sign.

           

            We argued that the magnitude of the average velocity is not the average speed. Notice the difference when discussing instantaneous values. The magnitude of the instantaneous velocity is the instantaneous speed. In an infinitesimal time interval, the magnitude of the displacement is equal to the distance traveled by the particle.

Acceleration

            When the velocity of a particle changes with time, the particle is said to be accelerating. The average acceleration a–x of the particle is defined as the change in velocity ∆vx divided by the time interval ∆t during which that change occurs. The instantaneous acceleration equals the derivative of the velocity

with respect to time.

            As with velocity, when the motion being analyzed is one-dimensional, we can use

positive and negative signs to indicate the direction of the acceleration.

            Keep in mind that negative acceleration does not necessarily mean that an object is slowing down. If the acceleration is negative, and the velocity is negative, the object is speeding up!  When the object’s velocity and acceleration are in the same direction, the object is speeding up. On the other hand, when the object’s velocity and acceleration are in opposite directions, the object is slowing down. If the car is slowing down, a force must be pulling in the direction opposite to its velocity.

Freely Falling Objects

The Italian Galileo Galilei (1564–1642) originated our present-day ideas concerning falling objects. There is a legend that he demonstrated the behavior of falling objects by observing that two different weights dropped simultaneously from the Leaning Tower of Pisa hit the ground at approximately the same time. Although there is some doubt that he carried out this particular experiment, it is well established that Galileo

performed many experiments on objects moving on inclined planes. In his experiments he rolled balls down a slight incline and measured the distances they covered in successive time intervals. The purpose of the incline was to reduce the acceleration; with the acceleration reduced, Galileo was able to make accurate measurements of the time intervals. By gradually increasing the slope of the incline, he was finally able to draw conclusions about freely falling objects because a freely falling ball is equivalent to a ball moving down a vertical incline.

            When we use the expression freely falling object, we do not necessarily refer to an object dropped from rest. A freely falling object is any object moving freely under the influence of gravity alone, regardless of its initial motion. Objects thrown upward or downward and those released from rest are all falling freely once they are released. Any freely falling object experiences an acceleration directed downward, regardless of its initial motion.