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Vectors r of t and r of t plus delta t are shown as red arrows in x y coordinate system. Both vectors start at the origin. Vector delta r points from the head of vector r of t to the head of vector r of t plus delta t.
A particle moves along a path given by the gray line. In the limit as Δ t approaches zero, the velocity vector becomes tangent to the path of the particle.

[link] can also be written in terms of the components of v ( t ) . Since

r ( t ) = x ( t ) i ^ + y ( t ) j ^ + z ( t ) k ^ ,

we can write

v ( t ) = v x ( t ) i ^ + v y ( t ) j ^ + v z ( t ) k ^

where

v x ( t ) = d x ( t ) d t , v y ( t ) = d y ( t ) d t , v z ( t ) = d z ( t ) d t .

If only the average velocity is of concern, we have the vector equivalent of the one-dimensional average velocity for two and three dimensions:

v avg = r ( t 2 ) r ( t 1 ) t 2 t 1 .

Calculating the velocity vector

The position function of a particle is r ( t ) = 2.0 t 2 i ^ + ( 2.0 + 3.0 t ) j ^ + 5.0 t k ^ m . (a) What is the instantaneous velocity and speed at t = 2.0 s? (b) What is the average velocity between 1.0 s and 3.0 s?

Solution

Using [link] and [link] , and taking the derivative of the position function with respect to time, we find

(a) v ( t ) = d r ( t ) d t = 4.0 t i ^ + 3.0 j ^ + 5.0 k ^ m/s

v ( 2.0 s ) = 8.0 i ^ + 3.0 j ^ + 5.0 k ^ m/s

Speed | v ( 2.0 s ) | = 8 2 + 3 2 + 5 2 = 9.9 m/s .

(b) From [link] ,
v avg = r ( t 2 ) r ( t 1 ) t 2 t 1 = r ( 3.0 s ) r ( 1.0 s ) 3.0 s 1.0 s = ( 18 i ^ + 11 j ^ + 15 k ^ ) m ( 2 i ^ + 5 j ^ + 5 k ^ ) m 2.0 s = ( 16 i ^ + 6 j ^ + 10 k ^ ) m 2.0 s = 8.0 i ^ + 3.0 j ^ + 5.0 k ^ m/s .

Significance

We see the average velocity is the same as the instantaneous velocity at t = 2.0 s, as a result of the velocity function being linear. This need not be the case in general. In fact, most of the time, instantaneous and average velocities are not the same.

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Check Your Understanding The position function of a particle is r ( t ) = 3.0 t 3 i ^ + 4.0 j ^ . (a) What is the instantaneous velocity at t = 3 s? (b) Is the average velocity between 2 s and 4 s equal to the instantaneous velocity at t = 3 s?

(a) Taking the derivative with respect to time of the position function, we have v ( t ) = 9.0 t 2 i ^ and v (3.0s) = 81.0 i ^ m/s . (b) Since the velocity function is nonlinear, we suspect the average velocity is not equal to the instantaneous velocity. We check this and find
v avg = r ( t 2 ) r ( t 1 ) t 2 t 1 = r ( 4.0 s ) r ( 2.0 s ) 4.0 s 2.0 s = ( 144.0 i ^ 36.0 i ^ ) m 2.0 s = 54.0 i ^ m/s ,
which is different from v (3.0s) = 81.0 i ^ m/s .

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The independence of perpendicular motions

When we look at the three-dimensional equations for position and velocity written in unit vector notation, [link] and [link] , we see the components of these equations are separate and unique functions of time that do not depend on one another. Motion along the x direction has no part of its motion along the y and z directions, and similarly for the other two coordinate axes. Thus, the motion of an object in two or three dimensions can be divided into separate, independent motions along the perpendicular axes of the coordinate system in which the motion takes place.

To illustrate this concept with respect to displacement, consider a woman walking from point A to point B in a city with square blocks. The woman taking the path from A to B may walk east for so many blocks and then north (two perpendicular directions) for another set of blocks to arrive at B . How far she walks east is affected only by her motion eastward. Similarly, how far she walks north is affected only by her motion northward.

Independence of motion

In the kinematic description of motion, we are able to treat the horizontal and vertical components of motion separately. In many cases, motion in the horizontal direction does not affect motion in the vertical direction, and vice versa.

Questions & Answers

a particle projected from origin moving on x-y plane passes through P & Q having consituents (9,7) , (18,4) respectively.find eq. of trajectry.
rahul Reply
definition of inertia
philip Reply
the reluctance of a body to start moving when it is at rest and to stop moving when it is in motion
charles
An inherent property by virtue of which the body remains in its pure state or initial state
Kushal
why current is not a vector quantity , whereas it have magnitude as well as direction.
Aniket Reply
why
daniel
the flow of current is not current
fitzgerald
bcoz it doesn't satisfy the algabric laws of vectors
Shiekh
The Electric current can be defined as the dot product of the current density and the differential cross-sectional area vector : ... So the electric current is a scalar quantity . Scalars are related to tensors by the fact that a scalar is a tensor of order or rank zero .
Kushal
what is binomial theorem
Tollum Reply
hello are you ready to ask aquestion?
Saadaq Reply
what is binary operations
Tollum
What is the formula to calculat parallel forces that acts in opposite direction?
Martan Reply
position, velocity and acceleration of vector
Manuel Reply
hi
peter
hi
daniel
hi
Vedisha
*a plane flies with a velocity of 1000km/hr in a direction North60degree east.find it effective velocity in the easterly and northerly direction.*
imam
hello
Lydia
hello Lydia.
Sackson
What is momentum
isijola
hello
Saadaq
A rail way truck of mass 2400kg is hung onto a stationary trunk on a level track and collides with it at 4.7m|s. After collision the two trunk move together with a common speed of 1.2m|s. Calculate the mass of the stationary trunk
Ekuri Reply
I need the solving for this question
philip
is the eye the same like the camera
EDWIN Reply
I can't understand
Suraia
same here please
Josh
I think the question is that ,,, the working principal of eye and camera same or not?
Sardar
yes i think is same as the camera
muhammad
what are the dimensions of surface tension
samsfavor
why is the "_" sign used for a wave to the right instead of to the left?
MUNGWA Reply
why classical mechanics is necessary for graduate students?
khyam Reply
classical mechanics?
Victor
principle of superposition?
Naveen Reply
principle of superposition allows us to find the electric field on a charge by finding the x and y components
Kidus
Two Masses,m and 2m,approach each along a path at right angles to each other .After collision,they stick together and move off at 2m/s at angle 37° to the original direction of the mass m. What where the initial speeds of the two particles
MB
2m & m initial velocity 1.8m/s & 4.8m/s respectively,apply conservation of linear momentum in two perpendicular directions.
Shubhrant
A body on circular orbit makes an angular displacement given by teta(t)=2(t)+5(t)+5.if time t is in seconds calculate the angular velocity at t=2s
MB
2+5+0=7sec differentiate above equation w.r.t time, as angular velocity is rate of change of angular displacement.
Shubhrant
Ok i got a question I'm not asking how gravity works. I would like to know why gravity works. like why is gravity the way it is. What is the true nature of gravity?
Daniel Reply
gravity pulls towards a mass...like every object is pulled towards earth
Ashok
An automobile traveling with an initial velocity of 25m/s is accelerated to 35m/s in 6s,the wheel of the automobile is 80cm in diameter. find * The angular acceleration
Goodness Reply
(10/6) ÷0.4=4.167 per sec
Shubhrant
what is the formula for pressure?
Goodness Reply
force/area
Kidus
force is newtom
Kidus
and area is meter squared
Kidus
so in SI units pressure is N/m^2
Kidus
In customary United States units pressure is lb/in^2. pound per square inch
Kidus
Practice Key Terms 3

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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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