1 Chapter 2: Motion along a Straight Line. 2 Displacement, Time, Velocity.

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Chapter 2: Motion along a Straight Line

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Displacement, Time, Velocity

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One-Dimensional Motion

The area of physics that we focus on is called mechanics: the study of the relationships between force, matter and motion

For now we focus on kinematics: the language used to describe motion

Later we will study dynamics: the relationship between motion and its causes (forces)

Simplest kind of motion: 1-D motion (along a straight line) A particle is a model of moving body in absence of effects

such as change of shape and rotation Velocity and acceleration are physical quantities to describe

the motion of particle Velocity and acceleration are vectors

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Position and Displacement

Motion is purely translational, when there is no rotation involved. Any object that is undergoing purely translational motion can be described as a point particle (an object with no size).

The position of a particle is a vector that points from the origin of a coordinate system to the location of the particle

The displacement of a particle over a given time interval is a vector that points from its initial position to its final position. It is the change in position of the particle.

To study the motion, we need coordinate system

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Position and Displacement

Motion of the “particle” on the dragster can be described in terms of the change in particle’s position over time interval

Displacement of particle is a vector pointing from P1 to P2

along the x-axis

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Average Velocity

Average velocity during this time interval is a vector quantity whose x-component is the change in x divided by the time interval

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Average Velocity

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Average velocity is positive when during the time interval coordinate x increased and particle moved in the positive direction

If particle moves in negative x-direction during time interval, average velocity is negative

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X-t Graph

This graph is pictorial way to represent how particle position changes in time

Average velocity depends only on total displacement x, not on the details of what happens during time interval t

The average speed of a particle is scalar quantity that is equal to the total distance traveled divided by the total time elapsed.

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Average Velocity

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Instantaneous Velocity

Instantaneous velocity of a particle is a vector equal to the limit of the average velocity as the time interval approaches zero. It equals the instantaneous rate of change of position with respect to time.

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Instantaneous Velocity

On a graph of position as a function of time for one-dimensional motion, the instantaneous velocity at a point is equal to the slope of the tangent to the curve at that point.

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Instantaneous Velocity

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Instantaneous Velocity

Concept QuestionThe graph shows position versus time for a particle undergoing 1-D motion.

At which point(s) is the velocity vx positive?

At which point(s) is the velocity negative?

At which point(s) is the velocity zero?

At which point is speed the greatest?

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Acceleration

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Acceleration

If the velocity of an object is changing with time, then the object is undergoing an acceleration.

Acceleration is a measure of the rate of change of velocity with respect to time.

Acceleration is a vector quantity. In straight-line motion its only non-zero component is along

the axis along which the motion takes place.

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Average Acceleration

Average Acceleration over a given time interval is defined as the change in velocity divided by the change in time.

In SI units acceleration has units of m/s2.

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Instantaneous Acceleration

Instantaneous acceleration of an object is obtained by letting the time interval in the above definition of average acceleration become very small. Specifically, the instantaneous acceleration is the limit of the average acceleration as the time interval approaches zero:

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Acceleration of Graphs

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Acceleration of Graphs

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Acceleration of Graphs

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Constant Acceleration Motion

In the special case of constant acceleration: the velocity will be a linear function of time, and the position will be a quadratic function of time. For this type of motion, the relationships between position,

velocity and acceleration take on the simple forms :

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Constant Acceleration Motion

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Constant Acceleration Motion

Position of a particle moving with constant acceleration

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Constant Acceleration Motion

Relationship between position of a particle moving with constant acceleration, and velocity and acceleration itself:

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Freely Falling Bodies

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Freely Falling Bodies

Example of motion with constant acceleration is acceleration of a body falling under influence of the earth’s gravitation

All bodies at a particular location fall with the same downward acceleration, regardless of their size and weight

Idealized motion free fall: we neglect earth rotation, decrease of acceleration with decreasing altitude, air effects

Galileo Galilei1564 - 1642

Aristotle   384 - 322 B.C.E.

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Freely Falling Bodies

The constant acceleration of a freely falling body is called acceleration due to gravity, g

Approximate value near earth’s surface g = 9.8 m/s2 = 980 cm/s2 = 32 ft/s2

g is the magnitude of a vector, it is always positive number

Exact g value varies with location

Acceleration due to gravity Near the sun: 270 m/s2

Near the moon: 1.6 m/s2