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Solution

(a) The origin of the coordinate system is at the top of the hill with y- axis vertically upward and the x- axis horizontal. By looking at the trajectory of the skier, the x- component of the acceleration is positive and the y- component is negative. Since the angle is 15 ° down the slope, we find

a x = ( 2.1 m/ s 2 ) cos ( 15 ° ) = 2.0 m/ s 2
a y = ( −2.1 m/ s 2 ) sin 15 ° = −0.54 m/ s 2 .

Inserting the initial position and velocity into [link] and [link] for x , we have

x ( t ) = 75.0 m + ( 4.1 m/s ) t + 1 2 ( 2.0 m/ s 2 ) t 2
v x ( t ) = 4.1 m/s + ( 2.0 m/ s 2 ) t .

For y , we have

y ( t ) = −50.0 m + ( −1.1 m/s ) t + 1 2 ( −0.54 m/ s 2 ) t 2
v y ( t ) = −1.1 m/s + ( −0.54 m/ s 2 ) t .

(b) Now that we have the equations of motion for x and y as functions of time, we can evaluate them at t = 10.0 s:

x ( 10.0 s ) = 75.0 m + ( 4.1 m/ s 2 ) ( 10.0 s ) + 1 2 ( 2.0 m/ s 2 ) ( 10.0 s ) 2 = 216.0 m
v x ( 10.0 s ) = 4.1 m/s + ( 2.0 m/ s 2 ) ( 10.0 s ) = 24.1 m /s
y ( 10.0 s ) = −50.0 m + ( −1.1 m/s ) ( 10.0 s ) + 1 2 ( −0.54 m/ s 2 ) ( 10.0 s ) 2 = −88.0 m
v y ( 10.0 s ) = −1.1 m/s + ( −0.54 m/ s 2 ) ( 10.0 s ) = −6.5 m/s .

The position and velocity at t = 10.0 s are, finally,

r ( 10.0 s ) = ( 216.0 i ^ 88.0 j ^ ) m
v ( 10.0 s ) = ( 24.1 i ^ 6.5 j ^ ) m/s .

The magnitude of the velocity of the skier at 10.0 s is 25 m/s, which is 60 mi/h.

Significance

It is useful to know that, given the initial conditions of position, velocity, and acceleration of an object, we can find the position, velocity, and acceleration at any later time.

With [link] through [link] we have completed the set of expressions for the position, velocity, and acceleration of an object moving in two or three dimensions. If the trajectories of the objects look something like the “Red Arrows” in the opening picture for the chapter, then the expressions for the position, velocity, and acceleration can be quite complicated. In the sections to follow we examine two special cases of motion in two and three dimensions by looking at projectile motion and circular motion.

At this University of Colorado Boulder website , you can explore the position velocity and acceleration of a ladybug with an interactive simulation that allows you to change these parameters.

Summary

  • In two and three dimensions, the acceleration vector can have an arbitrary direction and does not necessarily point along a given component of the velocity.
  • The instantaneous acceleration is produced by a change in velocity taken over a very short (infinitesimal) time period. Instantaneous acceleration is a vector in two or three dimensions. It is found by taking the derivative of the velocity function with respect to time.
  • In three dimensions, acceleration a ( t ) can be written as a vector sum of the one-dimensional accelerations a x ( t ) , a y ( t ) , and a z ( t ) along the x- , y -, and z- axes.
  • The kinematic equations for constant acceleration can be written as the vector sum of the constant acceleration equations in the x , y , and z directions.

Conceptual questions

If the position function of a particle is a linear function of time, what can be said about its acceleration?

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If an object has a constant x -component of the velocity and suddenly experiences an acceleration in the y direction, does the x- component of its velocity change?

No, motions in perpendicular directions are independent.

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If an object has a constant x- component of velocity and suddenly experiences an acceleration at an angle of 70 ° in the x direction, does the x- component of velocity change?

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Problems

The position of a particle is r ( t ) = ( 3.0 t 2 i ^ + 5.0 j ^ 6.0 t k ^ ) m . (a) Determine its velocity and acceleration as functions of time. (b) What are its velocity and acceleration at time t = 0?

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A particle’s acceleration is ( 4.0 i ^ + 3.0 j ^ ) m/ s 2 . At t = 0, its position and velocity are zero. (a) What are the particle’s position and velocity as functions of time? (b) Find the equation of the path of the particle. Draw the x- and y- axes and sketch the trajectory of the particle.

a. v ( t ) = ( 4.0 t i ^ + 3.0 t j ^ ) m/s , r ( t ) = ( 2.0 t 2 i ^ + 3 2 t 2 j ^ ) m ,
b. x ( t ) = 2.0 t 2 m, y ( t ) = 3 2 t 2 m, t 2 = x 2 y = 3 4 x
A graph of the linear function y equals 3 quarters x. The graph is a straight, positive slope line through the origin.

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A boat leaves the dock at t = 0 and heads out into a lake with an acceleration of 2.0 m/ s 2 i ^ . A strong wind is pushing the boat, giving it an additional velocity of 2.0 m/s i ^ + 1.0 m/s j ^ . (a) What is the velocity of the boat at t = 10 s? (b) What is the position of the boat at t = 10s? Draw a sketch of the boat’s trajectory and position at t = 10 s, showing the x- and y -axes.

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The position of a particle for t >0 is given by r ( t ) = ( 3.0 t 2 i ^ 7.0 t 3 j ^ 5.0 t −2 k ^ ) m . (a) What is the velocity as a function of time? (b) What is the acceleration as a function of time? (c) What is the particle’s velocity at t = 2.0 s? (d) What is its speed at t = 1.0 s and t = 3.0 s? (e) What is the average velocity between t = 1.0 s and t = 2.0 s?

a. v ( t ) = ( 6.0 t i ^ 21.0 t 2 j ^ + 10.0 t −3 k ^ ) m/s ,
b. a ( t ) = ( 6.0 i ^ 42.0 t j ^ 30 t −4 k ^ ) m/ s 2 ,
c. v ( 2.0 s ) = ( 12.0 i ^ 84.0 j ^ + 1.25 k ^ ) m/s ,
d. v ( 1.0 s ) = 6.0 i ^ 21.0 j ^ + 10.0 k ^ m/s , | v ( 1.0 s ) | = 24.0 m/s
v ( 3.0 s ) = 18.0 i ^ 189.0 j ^ + 0.37 k ^ m/s , | v ( 3.0 s ) | = 199.0 m/s ,
e. r ( t ) = ( 3.0 t 2 i ^ 7.0 t 3 j ^ 5.0 t −2 k ^ ) cm
v avg = 9.0 i ^ 49.0 j ^ 6.3 k ^ m/s

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The acceleration of a particle is a constant. At t = 0 the velocity of the particle is ( 10 i ^ + 20 j ^ ) m/s . At t = 4 s the velocity is 10 j ^ m/s . (a) What is the particle’s acceleration? (b) How do the position and velocity vary with time? Assume the particle is initially at the origin.

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A particle has a position function r ( t ) = cos ( 1.0 t ) i ^ + sin ( 1.0 t ) j ^ + t k ^ , where the arguments of the cosine and sine functions are in radians. (a) What is the velocity vector? (b) What is the acceleration vector?

a. v ( t ) = −sin ( 1.0 t ) i ^ + cos ( 1.0 t ) j ^ + k ^ , b. a ( t ) = −cos ( 1.0 t ) i ^ sin ( 1.0 t ) j ^

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A Lockheed Martin F-35 II Lighting jet takes off from an aircraft carrier with a runway length of 90 m and a takeoff speed 70 m/s at the end of the runway. Jets are catapulted into airspace from the deck of an aircraft carrier with two sources of propulsion: the jet propulsion and the catapult. At the point of leaving the deck of the aircraft carrier, the F-35’s acceleration decreases to a constant acceleration of 5.0 m/ s 2 at 30 ° with respect to the horizontal. (a) What is the initial acceleration of the F-35 on the deck of the aircraft carrier to make it airborne? (b) Write the position and velocity of the F-35 in unit vector notation from the point it leaves the deck of the aircraft carrier. (c) At what altitude is the fighter 5.0 s after it leaves the deck of the aircraft carrier? (d) What is its velocity and speed at this time? (e) How far has it traveled horizontally?

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Questions & Answers

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
who is Newton?
John Reply
scientist
Jeevan
a scientist
Peter
that discovered law of motion
Peter
ok
John
but who is Isaac newton?
John
a postmodernist would say that he did not discover them, he made them up and they're not actually a reality in itself, but a mere construct by which we decided to observe the word around us
elo
how?
Qhoshe
Besides his work on universal gravitation (gravity), Newton developed the 3 laws of motion which form the basic principles of modern physics. His discovery of calculus led the way to more powerful methods of solving mathematical problems. His work in optics included the study of white light and
Daniel
and the color spectrum
Daniel
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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