Chapter 2 Motion in one dimension

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Chapter 2: Motion in one dimension

The study of motion and of physical concepts such as

force and mass is called dynamics.

The part of dynamics that describes motion without

regard to its causes is called kinematics.

The purpose of this chapter is to describe motion using

the concepts of displacement, velocity, and

acceleration. For the sake of simplicity, we begin with

the study of 1-dimensional motion.

1) Displacement

Motion involves the displacement of an object from one place in space and time to another.

Describing the motion requires some convenient coordinate system and a specified origin. A frame

of reference is a choice of coordinate axes that defines the starting point for measuring any quantity.

Ex: Consider a body moving in 1-dimension; a train traveling down a straight railroad track:

The x-coordinate of the train at any time describes its position in space.

The displacement of an object is defined as its change in position, and is given by:

SI unit: meter (m)

where the initial position of the object is labeled and the final position is

.

Note: The displacement of an object is not the same as the distance it travels: when you toss a

ball 1 m up and you catch it; the displacement is zero but the distance covered by the ball is 2 m.

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2) Velocity

2.1) Speed vs velocity

In day-to-day usage, the terms speed and velocity are interchangeable. In physics, however, there’s a

clear distinction between them: Speed is a scalar quantity, having only magnitude, while velocity is

a vector, having both magnitude and direction.

2.2) Average speed

The average speed of an object over a given time interval is defined as the total distance traveled

divided by the total time elapsed:

Average speed is always positive.

2.3) Average velocity

a) Definition

The average velocity during a time interval t is the displacement divided by t :

The average velocity of an object in one dimension can be either positive or negative, depending

on the sign of the displacement.

Example 1:

If you run from x = 0 m to x = 25 m and back to your starting point in a time interval of 5 s.

Compare your average speed with your average velocity.

2.4) Instantaneous velocity

Average velocity doesn’t take into account the details of what happens during an interval of time. To

do so, we use the concept of instantaneous velocity. The instantaneous velocity is the limit of the

average velocity as the time interval becomes infinitesimally small:

SI unit: (m/s)

1 m/s = 3.6 km/h.

SI unit: (m/s)

SI unit: (m/s)

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3) Acceleration

The changing of an object’s velocity with time is called acceleration.

The instantaneous acceleration is given by:

4) One dimensional Motion with constant velocity

For a 1-D motion with constant velocity, the kinematic equation of motion is given by:

5) One dimensional Motion with constant acceleration

For a 1-D motion with constant acceleration, the Kinematics equations are:

We can also use:

Example 2:

A race car starting from rest accelerates at a constant rate of 5 m/s².

1) What is the velocity of the car after it has traveled 30.5 m?

2) How much time has elapsed?

3) Calculate the average velocity two different ways.

Example 3:

A typical jetliner lands at a speed of 71.5 m/s and decelerates at the rate of 4.47 m/s². If the plane

travels at a constant speed of 71.5 m/s for 1.00 s after landing before applying the brakes, what is the

total displacement of the aircraft between touchdown on the runway and coming to rest?

6) Freely falling objects

A freely falling object is any object moving freely under the influence of gravity alone,

regardless of its initial motion. Ex: Objects thrown upward, downward or released from rest.

If we neglect air resistance and assume that the free-fall acceleration doesn’t vary with altitude over

short vertical distances, then the motion of a freely falling object is the same as motion in one

dimension under constant acceleration. If we choose the up-direction as the +y-direction:

SI unit: (m/s2

)

for constant a

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The kinematics equations of motion with the y-variable are:

Example 4:

A stone is thrown from the top of a building with an initial velocity of

straight upward, at an initial height of above the ground.

The stone just misses the edge of the roof on its way down, as shown

in the figure. Neglect air drag. Determine:

1) the time needed for the stone to reach its maximum height,

2) the maximum

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