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An x y z coordinate system is shown. All the axes show distance in meters and run from -50 to 50 meters. A series of 10 red dots are shown, with the sixth dot is labeled as t = 6 s and the tenth as t = 10 s. The red series of dots starts at the origin and curves upward (both y and z increasing with time). Vertical dashed lines connect the red dots to a series of blue dots in the x y plane. The blue dots are all in the first quadrant (positive x and y). The dots are regularly spaced along the y coordinate, while the x coordinate starts at 0, increases, reaches a maximum of x = 25 m at t = 5, and then decreases back to x = 0 at t 10 s.
The particle starts at point ( x , y , z ) = (0, 0, 0) with position vector r = 0 . The projection of the trajectory onto the xy- plane is shown. The values of y and z increase linearly as a function of time, whereas x has a turning point at t = 5 s and 25 m, when it reverses direction. At this point, the x component of the velocity becomes negative. At t = 10 s, the particle is back to 0 m in the x direction.

Significance

By graphing the trajectory of the particle, we can better understand its motion, given by the numerical results of the kinematic equations.

Check Your Understanding Suppose the acceleration function has the form a ( t ) = a i ^ + b j ^ + c k ^ m/ s 2 , where a, b, and c are constants. What can be said about the functional form of the velocity function?

The acceleration vector is constant and doesn’t change with time. If a, b , and c are not zero, then the velocity function must be linear in time. We have v ( t ) = a d t = ( a i ^ + b j ^ + c k ^ ) d t = ( a i ^ + b j ^ + c k ^ ) t m/s , since taking the derivative of the velocity function produces a ( t ) . If any of the components of the acceleration are zero, then that component of the velocity would be a constant.

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Constant acceleration

Multidimensional motion with constant acceleration can be treated the same way as shown in the previous chapter for one-dimensional motion. Earlier we showed that three-dimensional motion is equivalent to three one-dimensional motions, each along an axis perpendicular to the others. To develop the relevant equations in each direction, let’s consider the two-dimensional problem of a particle moving in the xy plane with constant acceleration, ignoring the z -component for the moment. The acceleration vector is

a = a 0 x i ^ + a 0 y j ^ .

Each component of the motion has a separate set of equations similar to [link][link] of the previous chapter on one-dimensional motion. We show only the equations for position and velocity in the x - and y -directions. A similar set of kinematic equations could be written for motion in the z -direction:

x ( t ) = x 0 + ( v x ) avg t
v x ( t ) = v 0 x + a x t
x ( t ) = x 0 + v 0 x t + 1 2 a x t 2
v x 2 ( t ) = v 0 x 2 + 2 a x ( x x 0 )
y ( t ) = y 0 + ( v y ) avg t
v y ( t ) = v 0 y + a y t
y ( t ) = y 0 + v 0 y t + 1 2 a y t 2
v y 2 ( t ) = v 0 y 2 + 2 a y ( y y 0 ) .

Here the subscript 0 denotes the initial position or velocity. [link] to [link] can be substituted into [link] and [link] without the z -component to obtain the position vector and velocity vector as a function of time in two dimensions:

r ( t ) = x ( t ) i ^ + y ( t ) j ^ and v ( t ) = v x ( t ) i ^ + v y ( t ) j ^ .

The following example illustrates a practical use of the kinematic equations in two dimensions.

A skier

[link] shows a skier moving with an acceleration of 2.1 m/ s 2 down a slope of 15 ° at t = 0. With the origin of the coordinate system at the front of the lodge, her initial position and velocity are

r ( 0 ) = ( 75.0 i ^ 50.0 j ^ ) m

and

v ( 0 ) = ( 4.1 i ^ 1.1 j ^ ) m/s .

(a) What are the x- and y -components of the skier’s position and velocity as functions of time? (b) What are her position and velocity at t = 10.0 s?

An illustration of a skier in an x y coordinate system is shown. The skier is moving along a line that is 15 degrees below the horizontal x direction and has an acceleration of a = 2.1 meters per second squared also directed in his direction of motion. The acceleration is represented as a purple arrow.
A skier has an acceleration of 2.1 m/s 2 down a slope of 15 ° . The origin of the coordinate system is at the ski lodge.

Strategy

Since we are evaluating the components of the motion equations in the x and y directions, we need to find the components of the acceleration and put them into the kinematic equations. The components of the acceleration are found by referring to the coordinate system in [link] . Then, by inserting the components of the initial position and velocity into the motion equations, we can solve for her position and velocity at a later time t .

Questions & Answers

show whether or not the expression v^2= u^2 sin^2 d- 2gs is dimensionally constant
Oyetayo Reply
the period T of a pendulum depends on its mass m, length l and acceleration due to gravity g. using dimensional analysis, derive for T.
Oyetayo
what is physics
Satyabrata Reply
Physics is the tool humans use to understand the properties characteristics and interactions of where they live - the universe. Thus making laws and theories about the universe in a mathematical way derived from empirical results yielded in tons of experiments.
Jomari
This tool, the physics, also enhances their way of thinking. Evolving integrating and enhancing their critical logical rational and philosophical thinking since the greeks fired the first neurons of physics.
Jomari
nice
Satyabrata
Physics is also under the category of Physical Science which deals with the behavior and properties of physical quantities around us.
Angelo
Physical Science is under the category of Physics*... I prefer the most is Theoretical Physics where it deals with the philosophical view of our world.
Jomari
what is unit
Satyabrata Reply
Metric unit
Arzoodan
A unit is what comes after a number that gives a precise detail on what the number means. For example, 10 kilograms, 10 is the number while "kilogram" is the unit.
Angelo
there are also different types of units, but metric is the most widely used. It is called the SI system. Please research this on google.
Angelo
Unit? Bahay yon
Jomari
How did you get the value as Dcd=0.2Dab
ebewele Reply
Why as Dcd=0.2Dab? where are you got this formula?...
Arzoodan
since the distance Dcd=1.2 and the distance Dab=6.0 the ratio 1.2/6.0 gives the equation Dcd=0.2Dab
sunday
Well done.
Arzoodan
how do we add or deduct zero errors from result gotten using vernier calliper?
Aina Reply
how can i understand if the function are odd or even or neither odd or even
hamzaani
I don't get... do you mean positive or negative@hamzaani
Aina
Verner calliper is an old calculator
Antonio
Function is even if f(-x) =f(x)
Antonio
Function is odd if f(-x) = - f(x)
Antonio
what physical phenomena is resonance?
amrit Reply
is there any resonance in weight?
amrit
Resonance is due to vibrations and waves
Antonio
wait there is a chat here
dare
what is the difference between average velocity and magnitude of displacement
ibrahim
how velocity change with time
ibrahim
average velocity can be zero positive negative but magnitude of displacement is positive
amrit
if there is different displacement in same interval of time
amrit
Displacement can be zero, if you came back
Antonio
Displacement its a [L]
Antonio
Velocity its a vector
Antonio
Speed its the magnitude of velocity
Antonio
[Vt2-Vt1]/[t2-t1] = average velocity,another vector
Antonio
Distance, that and only that can't be negative, and is not a vector
Antonio
Distance its a metrical characteristic of the euclidean space
Antonio
Velocity change in time due a force acting (an acceleration)
Antonio
the change in velocity can be found using conservation of energy if the displacement is known
Jose
BEFORE = AFTER
Jose
kinetic energy + potential energy is equal to the kinetic energy after
Jose
the potential energy can be described as made times displacement times acceleration. I.e the work done on the object
Jose
mass*
Jose
from there make the final velocity the subject and solve
Jose
If its a conservative field
Antonio
So, no frictions in this case
Antonio
right
Jose
and if still conservative but force is in play then simply include work done by friction
Jose
Is not simple, is a very unknown force
Antonio
the vibration of a particle due to vibration of a similar particle close to it.
Aina
No, not so simple
Antonio
Frequency is involved
Antonio
mechanical wave?
Aina
All kind of waves, even in the sea
Antonio
will the LCR circut pure inductive if applied frequency becomes more than the natural frequency of AC circut? if yes , why?
Muskan Reply
LCR pure inductive? Is an nonsense
Antonio
what is photon
Elijah Reply
Photon is the effect of the Maxwell equations, it's the graviton of the electromagnetic field
Antonio
a particle representing a quantum of light or other electromagnetic radiation. A photon carries energy proportional to the radiation frequency but has zero rest mass.
Areej
Quantum it's not exact, its the elementary particle of electromagnetic field. Its not well clear if quantum theory its so, or if it's classical mechanics improved
Antonio
A photon is first and foremost a particle. And hence obeys Newtonian Mechanics. It is what visible light and other electromagnetic waves is made up of.
eli
No a photon has speed of light, and no mass, so is not Newtonian Mechanics
Antonio
photon is both a particle and a wave (It is the property called particle-wave duality). It is nearly massless, and travels at speed c. It interacts with and carries electromagnetic force.
Angelo
what are free vectors
sunday Reply
a vector hows point of action doesn't static . then vector can move bodily from one point to another point located on its original tragectory.
Anuj
A free vector its an element of an Affine Space
Antonio
Clay Matthews, a linebacker for the Green Bay Packers, can reach a speed of 10.0 m/s. At the start of a play, Matthews runs downfield at 45° with respect to the 50-yard line and covers 8.0 m in 1 s. He then runs straight down the field at 90° with respect to the 50-yard line for 12 m, with an elapsed time of 1.2 s. (a) What is Matthews’ final displacement from the start of the play? (b) What is his average velocity?
Macy Reply
Clay Matthews, a linebacker for the Green Bay Packers, can reach a speed of 10.0 m/s. At the start of a play, Matthews runs downfield at 45° with respect to the 50-yard line and covers 8.0 m in 1 s. He then runs straight down the field at 90° with respect to the 50-yard line for 12 m, with an elap
ibrahim
Very easy man
Antonio
how to find time moved by a mass on a spring
CHIDERA Reply
Maybe you mean frequency
Antonio
why hot soup is more tastier than cold soup?
Hamisi Reply
energy is involved
michael
hot soup is more energetic and thus enhances the flavor than a cold one.
Angelo
Its not Physics... Firstly, It falls under Anatomy. Your taste buds are the one to be blame not its coldness or hotness. Secondly, it depends on how the soup is done. Different soups possess different flavors and savors. If its on Physics, coldness of the soup will just bore you and if its hot...
Jomari
what is the importance of banking road in the circular path
Hamisi Reply
the coefficient of static friction of the tires and the pavement becomes less important because the angle of the banked curve helps friction to prevent slipping
Jose
an insect is at the end of the ring and the ring is rotating at an angular speed 'w' and it reaches to centre find its angular speed.
Bharani Reply
Angular speed is the rate at which an object changes its angle (measured) in radians, in a given time period. Angular speed has a magnitude (a value) only.  v represents the linear speed of a rotating object, r its radius, and ω its angular velocity in units of radians per unit of time, then v = rω
Lady
Angular speed = (final angle) - (initial angle) / time = change in position/time. ω = θ /t. ω = angular speed in radians/sec.
Lady
a boy through a ball with minimum velocity of 60 m/s and the ball reach ground 300 metre from him calculate angle of inclination
Emmanuel Reply
Practice Key Terms 1

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Source:  OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5
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