MOTION IN ONE DIMENSION

In this chapter we will talk about some concepts, they are; position, displacement, velocity, and acceleration. Then, using these concepts we study about motion of objects in one dimension with a constant acceleration.

2.1 Position, Velocity and Speed.

The motion of a particle is completely known if the particle’s position in space is known at all times. 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.

Position of the car at the various time
Position	t(s)	x(m)
A B C D E F	0 10 20 30 40 50	30 52 38 0 -37 -53

Figure and Table 2.1

We can easily determine the change in position of the car for various time intervals. So, 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

XF = final position

Xi = initial position

That’s 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 Ūx of a particle is defined as the particle’s displacement Dx divided by the time interval Dt 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:

The distinction between average velocity and average speed—average velocity is the displacement divided by the time interval, while average speed is the distance divided by the time interval.

2.2 Instantaneous Velocity and Speed

The instantaneous velocity Ūx equals the limiting value of the ratio Dx/Dt as Dt approaches zero. The instantaneous velocity can be positive, negative, or zero:

The instantaneous speed of a particle is equal to the magnitude of its instantaneous velocity. We argued that the magnitude of the average velocity is not the average speed. The magnitude of the instantaneous velocity is the instantaneous speed.

Example 2.3 Average and Instantaneous Velocity

Consider the following one-dimensional motions:

(A) A ball thrown directly upward rises to a highest point and falls back into the thrower’s hand. (B) A race car starts from rest and speeds up to 100 m/s.

(C) A spacecraft drifts through space at constant velocity. Are there any points in the motion of these objects at which the instantaneous velocity has the same value as the average velocity over the entire motion? If so, identify the point(s).

Solution :

(A) The average velocity for the thrown ball is zero because the ball returns to the starting point; thus its displacement is zero. (Remember that average velocity is defined as Dx/Dt.) There is one point at which the instantaneous velocity is zero—at the top of the motion.

(B) The car’s average velocity cannot be evaluated unambiguously with the information given, but it must be some value between 0 and 100 m/s. Because the car will have every instantaneous velocity between 0 and 100 m/s at some time during the interval, there must be some instant at which the instantaneous velocity is equal to the average velocity.

(C) Because the spacecraft’s instantaneous velocity is constant, its instantaneous velocity at any time and its average velocity over any time interval are the same.

2.3 Acceleration

When the velocity of a particle changes with time, the particle is said to be accelerating. The average acceleration ā_x of the particle is defined as the change in velocity DVx divided by the time interval Dt during which that change occurs:

In some situations, the value of the average acceleration may be different over different time intervals. It is therefore useful to define the instantaneous acceleration as the limit of the average acceleration as Dt approaches zero.

The instantaneous acceleration equals the derivative of the velocity with respect to time, which by definition is the slope of the velocity–time graph. We obtain the instantaneous acceleration:

2.4 Motion Diagrams

We use the car for objrct.

Car are equally spaced. The car is moving with constant positive velocity. Acceleration equals zero.

Car become farther apart as time increases. Velocity and acceleration are in the same direction. Acceleration is constant.Velocity is increasing. This shows positive acceleration and positive velocity.

Car become closer together as time increases. Acceleration and velocity are in opposite directions. Acceleration is constant. Velocity is decreasing. Positive velocity and negative acceleration.

Quick Quiz

• Known

(a) If a car is traveling eastward,its acceleration is eastward.

(b) If a car is slowing down, its acceleration must be negative.

(c) A particle with constant acceleration can never stop and stay stopped

• Asked: Which of the following is true?

• The Answer

c). If a particle with constant acceleration stops and its acceleration remains constant, it must begin to move again in the opposite direction. If it did not, the acceleration would change from its original constant value to zero.

2.5 One-Dimensional Motion with Constant Acceleration

When velocity changes by the same amount during each time interval, acceleration is constant. There are conections between displacement, time, velocity, and constant acceleration.

The average acceleration of a particle is defined as the ratio of the change

in its velocity Dvx divided by the time interval Dt during which that change occurs:

When an object moves with constant acceleration, instantaneous acceleration at a point a time

Vxf = Vxi t + ax t (Velocity as a function of time)

Because velocity at constant acceleration varies linearly, we can express the average velocity in any time interval as the arithmetic mean of the initial velocity Vxi and the final velocity Vxf :

V = ½ (Vxi + Vxf)

The following will clarify the relationship between displacement, time, velocity, and constant acceleration.

Xf - Xi = ½ (Vxi + Vxf) t

Xf = Xi + ½ (Vxi + Vxf) t (Position as a function of velocity and time)

= Xi + ½ (Vxi + Vxf + ax t) t

Xf = Xi + Vxi t + axt2 (Position as a function of time)

Finally, we can obtain an expression for the final velocity that does not contain

time as a variable by substituting the value of t

Vxf2 = Vxi2 + 2ax (Xf –Xi) (Velocity as a function of position)

That is, when the acceleration of a particle is zero, its velocity is constant and its position

changes linearly with time.

2.6 Freely Falling Objects

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. When we use the expression freely falling object, we do not necessarily refer to an object dropped from rest

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.

The free-fall acceleration g is constant over the range of motion, where g is equal to 9.80 m/s2.

When discussing the objects in free fall we can use above equations ,where for a we use the value of g given above. In addition, because the vertical motion, we will replace X with Y, and putting Xo in place Yo. We take Yo= 0 unless otherwise specified. It does not matter whether we choose the positive y direction upward or downward direction; important we should be consistent throughout the settlement of a matter.