4.2 Acceleration vector (Page 3/4)

University physics volume 1 Page 3 / 4

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 + \frac{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 + \frac{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) + \frac{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) + \frac{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,

\vec{r} (10.0 s) = (216.0 \hat{i} - 88.0 \hat{j}) m

\vec{v} (10.0 s) = (24.1 \hat{i} - 6.5 \hat{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 $\vec{a} (t)$ 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 $\vec{r} (t) = (3.0 t^{2} \hat{i} + 5.0 \hat{j} - 6.0 t \hat{k}) m .$ (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 $(4.0 \hat{i} + 3.0 \hat{j}) m/ s^{2} .$ 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. $\vec{v} (t) = (4.0 t \hat{i} + 3.0 t \hat{j}) m/s,$ $\vec{r} (t) = (2.0 t^{2} \hat{i} + \frac{3}{2} t^{2} \hat{j}) m$ ,
b. $x (t) = 2.0 t^{2} m, y (t) = \frac{3}{2} t^{2} m, t^{2} = \frac{x}{2} \Rightarrow y = \frac{3}{4} x$
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 $2.0 m/ s^{2} \hat{i} .$ A strong wind is pushing the boat, giving it an additional velocity of $2.0 m/s \hat{i} + 1.0 m/s \hat{j} .$ (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 $\vec{r} (t) = (3.0 t^{2} \hat{i} - 7.0 t^{3} \hat{j} - 5.0 t^{−2} \hat{k}) m .$ (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. $\vec{v} (t) = (6.0 t \hat{i} - 21.0 t^{2} \hat{j} + 10.0 t^{−3} \hat{k}) m/s$ ,
b. $\vec{a} (t) = (6.0 \hat{i} - 42.0 t \hat{j} - 30 t^{−4} \hat{k}) m/ s^{2}$ ,
c. $\vec{v} (2.0 s) = (12.0 \hat{i} - 84.0 \hat{j} + 1.25 \hat{k}) m/s$ ,
d. $\vec{v} (1.0 s) = 6.0 \hat{i} - 21.0 \hat{j} + 10.0 \hat{k} m/s, | \vec{v} (1.0 s) | = 24.0 m/s$
$\vec{v} (3.0 s) = 18.0 \hat{i} - 189.0 \hat{j} + 0.37 \hat{k} m/s,$ $| \vec{v} (3.0 s) | = 199.0 m/s$ ,
e. $\vec{r} (t) = (3.0 t^{2} \hat{i} - 7.0 t^{3} \hat{j} - 5.0 t^{−2} \hat{k}) cm$
$\begin{matrix} {\vec{v}}_{avg} & = 9.0 \hat{i} - 49.0 \hat{j} - 6.3 \hat{k} m/s \end{matrix}$

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The acceleration of a particle is a constant. At t = 0 the velocity of the particle is $(10 \hat{i} + 20 \hat{j}) m/s .$ At t = 4 s the velocity is $10 \hat{j} m/s .$ (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 $\vec{r} (t) = cos (1.0 t) \hat{i} + sin (1.0 t) \hat{j} + t \hat{k},$ 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. $\vec{v} (t) = −sin (1.0 t) \hat{i} + cos (1.0 t) \hat{j} + \hat{k}$ , b. $\vec{a} (t) = −cos (1.0 t) \hat{i} - sin (1.0 t) \hat{j}$

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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

Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.

Kate Reply

To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.

Muhammed

What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?

Kate Reply

what is the change in momentum of a body?

Eunice Reply

what is a capacitor?

Raymond Reply

Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.

Gautam

A motor travelling at 72km/m on sighting a stop sign applying the breaks such that under constant deaccelerate in the meters of 50 metres what is the magnitude of the accelerate

Maria Reply

please solve

Sharon

8m/s²

Aishat

What is Thermodynamics

Muordit

velocity can be 72 km/h in question. 72 km/h=20 m/s, v^2=2.a.x , 20^2=2.a.50, a=4 m/s^2.

Mehmet

A boat travels due east at a speed of 40meter per seconds across a river flowing due south at 30meter per seconds. what is the resultant speed of the boat

Saheed Reply

50 m/s due south east

Someone

which has a higher temperature, 1cup of boiling water or 1teapot of boiling water which can transfer more heat 1cup of boiling water or 1 teapot of boiling water explain your . answer

Ramon Reply

I believe temperature being an intensive property does not change for any amount of boiling water whereas heat being an extensive property changes with amount/size of the system.

Someone

Scratch that

Someone

temperature for any amount of water to boil at ntp is 100⁰C (it is a state function and and intensive property) and it depends both will give same amount of heat because the surface available for heat transfer is greater in case of the kettle as well as the heat stored in it but if you talk.....

Someone

about the amount of heat stored in the system then in that case since the mass of water in the kettle is greater so more energy is required to raise the temperature b/c more molecules of water are present in the kettle

Someone

definitely of physics

Haryormhidey Reply

how many start and codon

Esrael Reply

what is field

Felix Reply

physics, biology and chemistry this is my Field

ALIYU

field is a region of space under the influence of some physical properties

Collete

what is ogarnic chemistry

WISDOM Reply

determine the slope giving that 3y+ 2x-14=0

WISDOM

Another formula for Acceleration

Belty Reply

a=v/t. a=f/m a

IHUMA

innocent

Adah

pratica A on solution of hydro chloric acid,B is a solution containing 0.5000 mole ofsodium chlorid per dm³,put A in the burret and titrate 20.00 or 25.00cm³ portion of B using melting orange as the indicator. record the deside of your burret tabulate the burret reading and calculate the average volume of acid used?

Nassze Reply

how do lnternal energy measures

Esrael

Two bodies attract each other electrically. Do they both have to be charged? Answer the same question if the bodies repel one another.

JALLAH Reply

No. According to Isac Newtons law. this two bodies maybe you and the wall beside you. Attracting depends on the mass och each body and distance between them.

Dlovan

Are you really asking if two bodies have to be charged to be influenced by Coulombs Law?

Robert

like charges repel while unlike charges atttact

Raymond

What is specific heat capacity

Destiny Reply

Specific heat capacity is a measure of the amount of energy required to raise the temperature of a substance by one degree Celsius (or Kelvin). It is measured in Joules per kilogram per degree Celsius (J/kg°C).

AI-Robot

specific heat capacity is the amount of energy needed to raise the temperature of a substance by one degree Celsius or kelvin

ROKEEB

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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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