assignment 1_ kinematics in 2 and 3 dimensions

1/9/2014 Assignment 1: Kinematics in 2 and 3 Dimensions

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Assignment 1: Kinematics in 2 and 3 Dimensions

Due: 2:00am on Friday, September 5, 2014

To understand how points are awarded, read the Grading Policy for this assignment.

Arrow Hits Apple

An arrow is shot at an angle of above the horizontal. The arrow hits a tree a horizontal distance away, at the same height above the ground as it was shot. Use for the magnitude of

the acceleration due to gravity.

Part A

Find , the time that the arrow spends in the air.

Answer numerically in seconds, to two significant figures.

You did not open hints for this part.

ANSWER:

Suppose someone drops an apple from a vertical distance of 6.0 meters, directly above the point where the arrowhits the tree.

Part B

How long after the arrow was shot should the apple be dropped, in order for the arrow to pierce the apple as thearrow hits the tree?

Express your answer numerically in seconds, to two significant figures.


ANSWER:

Circular Launch

A ball is launched up a semicircular chute in such a way that at the top of the chute, just before it goes into free fall,the ball has a centripetal acceleration of magnitude 2 .

J

N # NT

0

B

= 0B

T

= 0E

T

#



Part A

How far from the bottom of the chute does the ball land?

Your answer for the distance the ball travels from the end of the chute should contain .


ANSWER:

Problem 3.50

Spiraling Up. It is common to see birds of prey rising upward on thermals. The paths they take may be spiral-like.You can model the spiral motion as uniform circular motion combined with a constant upward velocity. Assume abird completes a circle of radius 8.00 m every 5.00 s and rises vertically at a rate of 3.00 m/s.

Part A

Find the speed of the bird relative to the ground.

ANSWER:

Part B

Find the magnitude of the bird's acceleration.

ANSWER:

=

m/s



Part C

Find the direction of the bird's acceleration.

ANSWER:

Part D

Find the angle between the bird's velocity vector and the horizontal.

ANSWER:

Problem 3.69

Two tanks are engaged in a training exercise on level ground. The first tank fires a paint-filled training round with amuzzle speed of 236 at an angle 13.4 above the horizontal while advancing toward the second tank with a

speed of 13.5 relative to the ground. The second tank is retreating at a speed of 38.5 relative to the ground,

but is hit by the shell. You can ignore air resistance and assume the shell hits at the same height above ground fromwhich it was fired.

Part A

Find the distance between the tanks when the round was first fired.

Take free fall acceleration to be = 9.80 .

ANSWER:

Part B

Find the distance between the tanks at the time of impact.

Take free fall acceleration to be = 9.80 .

ANSWER:

NT

above the horizontal

NT

NT NT

# NT

m

# NT

m



Battleship Shells

A battleship simultaneously fires two shells toward two identical enemy ships. One shell hits ship A, which is closeby, and the other hits ship B, which is farther away. The two shells are fired at the same speed. Assume that airresistance is negligible and that the magnitude of the acceleration due to gravity is .

Note that after Part B the question setup changes slightly.

Part A

What shape is the trajectory (graph of y vs. x) of the shells?

ANSWER:

Part B

For two shells fired at the same speed which statement about the horizontal distance traveled is correct?


ANSWER:

Now, consider for the remaining parts of the question below that both shells are fired at an angle greater than 45degrees with respect to the horizontal. Remember that enemy ship A is closer than enemy ship B.

Part C

Which shell is fired at the larger angle?


ANSWER:

#

straight line

parabola

hyperbola

The shape cannot be determined.

The shell fired at a larger angle with respect to the horizontal lands farther away.

The shell fired at an angle closest to 45 degrees lands farther away.

The shell fired at a smaller angle with respect to the horizontal lands farther away.

The lighter shell lands farther away.



Part D

Which shell is launched with a greater vertical velocity, ?

ANSWER:

Part E

Which shell is launched with a greater horizontal velocity, ?

ANSWER:

Part F

Which shell reaches the greater maximum height?


ANSWER:

Part G

Which shell has the longest travel time (time elapsed between being fired and hitting the enemy ship)?


A

B

Both shells are fired at the same angle.

2

5

A

B

Both shells are launched with the same vertical velocity.

2

4

A

B

Both shells are launched with the same horizontal velocity.

A

B

Both shells reach the same maximum height.



ANSWER:

A Wild Ride

A car in a roller coaster moves along a track that consists of a sequence of ups and downs. Let the x axis be parallelto the ground and the positive y axis point upward. In the time interval from to s, the trajectory of the caralong a certain section of the track is given by

,

where is a positive dimensionless constant.

Part A

At is the roller coaster car ascending or descending?


ANSWER:

Part B

Derive a general expression for the speed of the car.

Express your answer in meters per second in terms of and .


ANSWER:

Part C

The roller coaster is designed according to safety regulations that prohibit the speed of the car from exceeding . Find the maximum value of allowed by these regulations.

Express your answer using two significant figures.

A

B

Both shells have the same travel time.

0 0

NT0 N N .

%

?

T

0

T

0

&

?

0 T

ascending

descending

2

0

= 2 NT

NT




ANSWER:

Shooting over a Hill

A projectile is fired with speed at an angle from the horizontal as shown in the figure .

Part A

Find the highest point in the trajectory, .

Express the highest point in terms of the magnitude of the acceleration due to gravity , the initial

velocity , and the angle .


ANSWER:

Part B

What is the range of the projectile, ?

Express the range in terms of , , and .


= NBY

2

J

#

2

J

=

2

J #



ANSWER:

Consider your advice to an artillery officer who has the following problem. From his current postition, he must shootover a hill of height at a target on the other side, which has the same elevation as his gun. He knows from hisaccurate map both the bearing and the distance to the target and also that the hill is halfway to the target. Toshoot as accurately as possible, he wants the projectile to just barely pass above the hill.

Part C

Find the angle above the horizontal at which the projectile should be fired.

Express your answer in terms of and .


ANSWER:

Part D

What is the initial speed?

Express in terms of , , and .


ANSWER:

Part E

Find , the flight time of the projectile.

Express the flight time in terms of and .


ANSWER:

=

J

= J

2

#

= 2

0

H

#

= 0H



Uniform Circular Motion

Learning Goal:

To find the velocity and acceleration vectors for uniform circular motion and to recognize that this acceleration is thecentripetal acceleration.

Suppose that a particle's position is given by the following expression:

.

Part A

Choose the answer that best completes the following sentence:The particle's motion at can be described by ____________.

ANSWER:

Part B

When does the particle first cross the negative x axis?

Express your answer in terms of some or all of the variables (Greek letter omega), , and .

ANSWER:

Now, consider the velocity and speed of the particle.

Part C

Find the particle's velocity as a function of time.

Express your answer using unit vectors (e.g., + , where and are functions of , , , and

).


ANSWER:

0 .

%

?

&

?

DPT[0 TJO[0%

?

&

?

0

an ellipse starting at time on the positive x axis

an ellipse starting at time on the positive y axis

a circle starting at time on the positive x axis

a circle starting at time on the positive y axis

0

0

0

0

[ R

= 04

%

?

&

?

[ 0

R

= 02



Part D

Find the speed of the particle at time .

Express your answer in terms of some or all of the variables , , and .


ANSWER:

Part E

Now find the acceleration of the particle.

Express your answer using unit vectors (e.g., + , where and are functions of , , , and

).

ANSWER:

Part F

Your calculation is actually a derivation of the centripetal acceleration. To see this, express the acceleration ofthe particle in terms of its position .

Express your answer in terms of some or all of the variables and .

ANSWER:

Part G

This question will be shown after you complete previous question(s).

Part H

Finally, express the magnitude of the particle's acceleration in terms of and using the expression you

obtained for the speed of the particle.

Express your answer in terms of one or both of the variables and .

ANSWER:

0

[ R

= 20 ] 0]2

%

?

&

?

[ 0

R

= 0

0.

0.

[

= 0

2

2



Curved Motion Diagram

The motion diagram shown in the figure represents a pendulum released from rest at an angle of 45 from thevertical. The dots in the motion diagram represent thepositions of the pendulum bob at eleven momentsseparated by equal time intervals. The green arrowsrepresent the average velocity between adjacent dots.Also given is a "compass rose" in which directions arelabeled with the letters of the alphabet.

Part A

What is the direction of the acceleration of the object at moment 5?

Enter the letter of the arrow with this direction from the compass rose in the figure. Type Z if theacceleration vector has zero length.


ANSWER:

Part B

What is the direction of the acceleration of the object at moments 0 and 10?

Enter the letters corresponding to the arrows with these directions from the compass rose in thefigure, separated by commas. Type Z if the acceleration vector has zero length.


ANSWER:

=

directions at time step 0, time step 10 =



A Canoe on a River

A canoe has a velocity of 0.440 southeast relative to the earth. The canoe is on a river that is flowing at 0.500

east relative to the earth.

Part A

Find the magnitude of the velocity of the canoe relative to the river.

Express your answer in meters per second.


ANSWER:

Part B

This question will be shown after you complete previous question(s).

Problem 3.34

The Ferris wheel in the figure , which rotates counterclockwise, is just starting up. At a given instant, a passenger onthe rim of the wheel and passing through the lowest point of his circular motion is moving at 3.00 m/s and is gainingspeed at a rate of .

NT

NT

2

DS

= 2DS

NT

NT



Part A

Find the magnitude of the passenger's acceleration at this instant.

ANSWER:

Part B

Find the direction of the passenger's acceleration at this instant.

ANSWER:

Problem 3.78

You are flying in a light airplane spotting traffic for a radio station. Your flight carries you due east above a highway.Landmarks below tell you that your speed is 52.0 relative to the ground and your air speed indicator also reads

52.0 . However, the nose of your airplane is pointed somewhat south of east and the station's weather person

tells you that the wind is blowing with speed 18.0 .

Part A

In which direction is the wind blowing?

Express your answer as an angle measured east of north.

ANSWER:

Score Summary:

Your score on this assignment is 0.0%.

NT

to the right of vertical

NT

NT

NT

east of north



You received 0 out of a possible total of 40 points.

assignment 1_ kinematics in 2 and 3 dimensions

Documents

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horizontal distance

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magnitude ofthe acceleration

horizontal ntnt nt