Motion#1

Chapter 2

Motion in One Dimension

04/13/23 IB Physics (IC NL) 1

Mechanics

Mechanics is a branch of physics concerned with the behavior of physical bodies when subjected to forces or displacements also it deals with matter and investigates energy.Mechanics is divided into three branches:1- Statics2- Kinematics3- Dynamics

In this slide we will discuss KINEMATICS.


Kinematics Kinematics is a part of mechanics that

studies motion in relationship to time. In kinematics, you are interested in the

description of motion Not concerned with the cause of the motion

(Dynamics) or bodies at rest (statics)


Motion A body or object is said to be in motion when it is

observed moving or changing position relative to a fixed reference as time passes.

Therefore motion is relative; example when you are in a car and you take the moving car as your reference then you see a building moving back relative to you.

Then there should exist a reference frame to your work and since we are studying this topic while on Earth, we suppose the Earth is our reference frame thus it is called the terrestrial frame.


Motion cont. Motion, to our level, is studied either according

to trajectory or according to speed. The trajectory is the path followed by the

center of mass of a moving object. Remark: our object could be: 1- either a solid body of considerable dimensions with a

center of mass where all the mass of the object is concentrated (also called center of gravity)

2- or a point mass with relative negligible dimensions and the mass is concentrated at the point itself.

Hint: don’t worry about this the question usually identifies the type of the object.


Motion according to trajectory.

Studying the motion according to trajectory is very wide so we are limiting it to a general one which is curvilinear and includes one of three:

Rectilinear: along a straight line Circular: along a circle Curvilinear along a curve.The first study is the rectilinear motion

that is motion in one dimension.04/13/23 IB Physics (IC NL) 6

Quantities in Motion Any motion involves three

concepts Displacement Velocity Acceleration

These concepts can be used to study objects in motion


Position Defined in terms

of a frame of reference One dimensional,

so generally the x- or y-axis

Defines a starting point for the motion


Displacement Defined as the change in position

or how much the object was displaced from its initial position.

f stands for final and i stands for initial May be represented as y if vertical Units are meters (m) in SI,

centimeters (cm) in cgs.

f ix x x


Displacements


Vector and Scalar Quantities Vector quantities need both

magnitude (size) and direction to completely describe them Generally denoted by boldfaced type

and an arrow over the letter + or – sign is also used in vector

representation. Scalar quantities are completely

described by magnitude only


Displacement Isn’t Distance The displacement of an object is

not the same as the distance it travels Example: Throw a ball straight up and

then catch it at the same point you released it

The distance is twice the height The displacement is zero


Speed The average speed of an object is

defined as the total distance traveled divided by the total time elapsed

Speed is a scalar quantity

total distanceAverage speed

total time

dv

t


Speed, cont Average speed totally ignores any

variations in the object’s actual motion during the trip

The total distance and the total time are all that is important

SI unit m.s-1


Velocity It takes time for an object to

undergo a displacement The average velocity is rate at

which the displacement occurs or simply the rate of change of displacement.

generally use a time interval, so let ti = 0

fi

averagefi

x xxv

t t t


Velocity continued Direction will be the same as the

direction of the displacement (time interval is always positive) + or - is sufficient

Units of velocity is also m.s-1 (SI) Other units may be given in a

problem, but generally will need to be converted to m.s-1 these like km/h


Speed vs. Velocity

Cars on both paths have the same average velocity since they had the same displacement in the same time interval

The car on the blue path will have a greater average speed since the distance it traveled is larger


Graphical Interpretation of Velocity Velocity can be determined from a

position-time graph Average velocity equals the slope

of the line joining the initial and final positions

An object moving with a constant velocity will have a graph that is a straight line.


Average Velocity, Constant

The straight line indicates constant velocity

The slope of the line is the value of the average velocity

Determine the equation of this straight line.

x= 2t - 20


Average Velocity, Non Constant

The motion in this type of motion is of non-constant velocity

The average velocity is the slope of the blue line joining two points


Instantaneous Velocity The limit of the average velocity as the

time interval becomes infinitesimally short, or as the time interval approaches zero

The instantaneous velocity indicates what is happening at every instant of time. It is a function of time.

The speedometer of a car indicates the instantaneous speed.

lim

0t

xv

t


Instantaneous Velocity on a Graph The slope of the line tangent to the

position-vs.-time graph is defined to be the instantaneous velocity at that time The instantaneous speed is defined as

the magnitude of the instantaneous velocity


Instantaneous Velocity on a Graph


Determine the instantaneous velocity at x = 15 m.About 2.5 ms-1

Motion according to speed In this part we study the motion of an

object depending on its velocity along a rectilinear trajectory.

Our study is limited now to:1- motion with constant speed or velocity

and called uniform motion (URM)2- motion with a constantly varying

velocity and called uniformly accelerated motion. (UARM)


Uniform rectilinear motion

URM is motion with constant velocity

The instantaneous velocities are always the same All the instantaneous velocities will

also equal the average velocity


Kinematic equations of URM

a = 0 v = constant x = vt + xo

Where: a is acceleration v is velocity x is displacement xo is initial displacement or distance

covered when started timing

and t is time 04/13/23 IB Physics (IC NL) 26

Acceleration Changing velocity (not-uniform)

means an acceleration is present Acceleration is the rate of change

of the velocity

Units is ms-² (SI)

fi

fi

v vva

t t t


Relationship Between Acceleration and Velocity

Uniform velocity (shown by red arrows maintaining the same size)

Acceleration equals zero


Relationship Between Velocity and Acceleration

Velocity and acceleration are in the same direction Acceleration is uniform (blue arrows maintain the same

length) Velocity is increasing uniformly (red arrows are getting

longer) Positive velocity and positive acceleration, thus Accelerated

motion. va>0


Relationship Between Velocity and Acceleration

Acceleration and velocity are in opposite directions

Acceleration is uniform (blue arrows maintain the same length)

Velocity is decreasing uniformly (red arrows are getting shorter)

Velocity is positive and acceleration is negative, thus decelerated motion. va<0


Kinematic Equations Used in situations with uniform

acceleration a = const. 1

2

2 2

12122

v u at

x vt u v t

x ut at

v u a x04/13/23 IB Physics (IC NL) 31

2

3

4

5

Kinematic equations cont. Equation 4 is derive when

plugging equation 2 in equation 3

Equation 5 is obtained when eliminating t from equation 2 and plug it in equation 3


Notes on the equations

2i f

averagev v

x v t t

Gives displacement as a function of velocity and time.

Use when you don’t know and aren’t asked for the acceleration.



Shows velocity as a function of acceleration and time

Use when you don’t know and aren’t asked to find the displacement

v u at


Graphical Interpretation of the Equation

u

u



Gives displacement as a function of time, velocity and acceleration

Use when you don’t know and aren’t asked to find the final velocity

21

2x ut at


Displacement-time graph


Acceleration graph When you are given a displacement-

time graph and the figure simulates a smiling face then the motion is accelerated. If it is a gloomy face then the motion is decelerated.



Gives velocity as a function of acceleration and displacement

Use when you don’t know and aren’t asked for the time

2 2 2v u a x


Graphical interpretation revisited

The area under the velocity-time graph represents the displacement.

Note: the area above the time axis is positive displacement where as the one below the axis Is negative.

Given the following diagram: the displacement is: x = 64 m


Graphical interpretation revisited

The area under the acceleration-time graph represents the change in velocity.

Given the following graph: What is the velocity relative to this motion: v = 45 ms-1


Problem-Solving Hints Read the problem Draw a diagram

Choose a coordinate system, label initial and final points, indicate a positive direction for velocities and accelerations

Label all quantities, be sure all the units are consistent Convert if necessary

Specify the type of motion according to a. Choose the appropriate kinematic equation


Problem-Solving Hints, cont Solve for the unknowns

You may have to solve two equations for two unknowns

Check your results Estimate and compare Check units


Galileo Galilei 1564 - 1642 Galileo formulated

the laws that govern the motion of objects in free fall

Also looked at: Inclined planes Relative motion Thermometers Pendulum


Free Fall All objects moving under the influence

of gravity only are said to be in free fall Free fall does not depend on the object’s

original motion All objects falling near the earth’s

surface fall with a constant acceleration The acceleration is called the

acceleration due to gravity, and indicated by g


Laws of free fall First law: in vacuum and in the same

place, all bodies, dense or light obey the same law of fall.

Second law: the trajectory followed by a freely falling object is the vertical, directed towards the center of the earth.

Third law: the motion of a freely falling object is uniformly accelerated of acceleration g.


Free fall on the moon


Hyperlink for previous page

Apollo 15 Hammer/Feather gravity demonstration


Motion in air Consequence: In air and in the same

place, dense bodies only and at the beginning of their fall obey the same law of fall as in vacuum.

When an object falls in air it is subjected to friction with air particles and to an opposing force called air resistance R.

R=kSV2

(go to the following slide) 04/13/23 IB Physics (IC NL) 49

Air resistance cont.

R = kSV2

R is air resistance k is the aerodynamic constant depending

on the shape of the object.S is cross-sectional area of object.V is velocity of object.


Aerodynamic shape and k.

For each shape there is a different k.

The last shape F has the least k so that air resistance is minimal thus this shape is recommended for the use in airplanes and it is called the aerodynamic shape.

Note: the object is moving to the left.


Terminal velocity When the air resistance reaches a

value that balances the weight of the object then

Ksv2=mg

This velocity is called the terminal velocity with which the object continues falling and it is constant.

ks

mgv


Acceleration due to Gravity Symbolized by g g = 9.80 ms-²

When estimating, use g 10 m.s-2

g is always directed downward toward the center of the earth

Ignoring air resistance and assuming g doesn’t vary with altitude over short vertical distances, free fall is constantly accelerated motion


Free Fall – an object dropped Initial velocity is

zero Let up be positive Use the kinematic

equations Generally use y

instead of x since vertical

Acceleration is g = -9.80 m.s-2

u= 0

a = g


Free Fall – an object thrown downward a = g = -9.80

m.s-2

Initial velocity 0 With upward

being positive, initial velocity will be negative


Free Fall -- object thrown upward Initial velocity is

upward, so positive The instantaneous

velocity at the maximum height is zero

a = g = -9.80 m.s-2 everywhere in the motion

u = 0


Thrown upward, cont. The motion may be symmetrical

Then tup = tdown

Then v = -u The motion may not be symmetrical

You can break the motion into various parts to understand but it is recommendable that you don’t do this and stick to one system of axes with specified directions that don’t change with the motion of the object.

Generally in initial direction of motion of the object.04/13/23 IB Physics (IC NL) 57

Non-symmetrical Free Fall

No need to divide the motion into segments

Object thrown up: Vertical axis directed

upwards Horizontal axis through

projection point O. Above O, displacement is

positive Below O, displacement is

negative.04/13/23 IB Physics (IC NL) 58

Combination Motions


Education

Motion#1