Motion Along a Straight Line - University of Texas at Dallasdavid.lary/CM/... · 8/31/2011 · Motion Along a Straight Line Wednesday, August 31, 11. Kinematics (from Greek κινεῖν,

Motion Along a Straight Line

Wednesday, August 31, 11

Kinematics (from Greek κινεῖν, kinein, to move) is the branch of classical mechanics that describes the motion of bodies (objects) and systems (groups of objects) without consideration of the forces that cause the motion.


Instantaneous Velocity

For most everyday objects in motion their velocity is continually changing. For this reason it is useful to keep track of the object’s instantaneous velocity

υ = limx→0

ΔxΔt

SI unit: meter per second, m/s


A safari and a chase






average speed =distance

elapsed timeelapsed time = distance

average speed

The Kingfisher takes a plunge




average speed


What equation are we going to use?




average speed



speed= distancetime




average speed



speed= distancetime




average speed



speed= distancetime

We want to know time, so what shall we do?




average speed



speed= distancetime

time= distancespeed

= 7m4m/s

=7s

We want to know time, so what shall we do?


Finding the acceleration


Motion with constant accelerationFor a wide range of idealized problems it is useful to consider the motion of particles with constant acceleration, e.g. falling bodies with no air resistance have the constant acceleration of gravity.

In the picture a strobe light begins to fire as the ball is dropped.

Notice how the space between images increases as the ball’s velocity grows.


v versus t plot for motion with constant accelerations


Motion with constant positive acceleration results in steadily increasing velocity.

Motion with constant acceleration




Constant acceleration equations of motion


Use Matlab to plot position as a function of time and to calculate average velocity

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

5

10

15

20

25

30

Time, t(s)

Posi

tion,

x(m

)

vav=13.7 m/s


Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley

Q2.1

A. point P. B. point Q. C. point R. D. point S.

E. not enough information in the graph to decide

This is the x–t graph of the motion of a particle. Of the four points P, Q, R, and S, the velocity vx is greatest (most positive) at



This is the x–t graph of the motion of a particle. Of the four points P, Q, R, and S, the velocity vx is greatest (most positive) at



A2.1



Q2.2This is the x–t graph of the motion of a particle. Of the four points P, Q, R, and S, the speed is greatest at





This is the x–t graph of the motion of a particle. Of the four points P, Q, R, and S, the speed is greatest at

A2.2





Q2.3This is the x–t graph of the motion of a particle. Of the four points P, Q, R, and S, the acceleration ax is greatest (most positive) at





This is the x–t graph of the motion of a particle. Of the four points P, Q, R, and S, the acceleration ax is greatest (most positive) at

A2.3





You toss a ball straight upward, in the positive direction. The ball falls freely under the influence of gravity.

At the highest point in the ball’s motion,

A. its velocity is zero and its acceleration is zero.

B. its velocity is zero and its acceleration is positive (upward).

C. its velocity is zero and its acceleration is negative (downward).

D. its velocity is positive (upward) and its acceleration is zero.

E. its velocity is positive (upward) and its acceleration is zero.

Q2.4



A. its velocity is zero and its acceleration is zero.

B. its velocity is zero and its acceleration is positive (upward).

C. its velocity is zero and its acceleration is negative (downward).

D. its velocity is positive (upward) and its acceleration is zero.

E. its velocity is positive (upward) and its acceleration is zero.

A2.4You toss a ball straight upward, in the positive direction. The ball falls freely under the influence of gravity.

At the highest point in the ball’s motion,



This is a motion diagram of an object moving along the x–direction with constant acceleration. The dots 1, 2, 3, … show the position of the object at equal time intervals ∆t.

Q2.5

xx = 0

5 4 3 2 1

A. vx < 0, ax = 0 B. vx < 0, ax > 0

C. vx < 0, ax < 0 D. vx > 0, ax > 0

E. vx > 0, ax < 0

At the time labeled 3, what are the signs of the object’s velocity vx and acceleration ax?



A2.5

xx = 0

5 4 3 2 1

A. vx < 0, ax = 0 B. vx < 0, ax > 0

C. vx < 0, ax < 0 D. vx > 0, ax > 0

E. vx > 0, ax < 0

At the time labeled 3, what are the signs of the object’s velocity vx and acceleration ax?





Which of the following vx–t graphs best matches the motion shown in the motion diagram?

Q2.6

xx = 0

5 4 3 2 1

t

vx

0

A.

t

vx

0

B.

t

vx

0

C.

t

vx

0

D.

t

vx

E.0




Which of the following vx–t graphs best matches the motion shown in the motion diagram?

A2.6

xx = 0

5 4 3 2 1

t

vx

0

A.

t

vx

0

B.

t

vx

0

C.

t

vx

0

D.

t

vx

E.0




Which of the following ax–t graphs best matches the motion shown in the motion diagram?

Q2.7

xx = 0

5 4 3 2 1

t

ax

0

A.

t

ax

0

B.

t

ax

0

C.

t

ax

0

D.

t

ax

E.0




Which of the following ax–t graphs best matches the motion shown in the motion diagram?

A2.7

xx = 0

5 4 3 2 1

t

ax

0

A.

t

ax

0

B.

t

ax

0

C.

t

ax

0

D.

t

ax

E.0



An object moves along the x–axis with constant acceleration. The initial position x0 is positive, the initial velocity is negative, and the acceleration is positive.

Which of the following vx–t graphs best describes this motion?

Q2.8

t

vx

0

A.

t

vx

0

B.

t

vx

0

C.

t

vx

0

D.

t

vx

E.0



An object moves along the x–axis with constant acceleration. The initial position x0 is positive, the initial velocity is negative, and the acceleration is positive.

Which of the following vx–t graphs best describes this motion?

A2.8

t

vx

0

A.

t

vx

0

B.

t

vx

0

C.

t

vx

0

D.

t

vx

E.0



This is the vx–t graph for an object moving along the x-axis.

Which of the following descriptions of the motion is most accurate?

Q2.9

A. The object is slowing down at a decreasing rate.

B. The object is slowing down at an increasing rate.

C. The object is speeding up at a decreasing rate.

D. The object is speeding up at an increasing rate.

E. The object’s speed is changing at a steady rate.



This is the vx–t graph for an object moving along the x-axis.

Which of the following descriptions of the motion is most accurate?

A2.9

A. The object is slowing down at a decreasing rate.

B. The object is slowing down at an increasing rate.

C. The object is speeding up at a decreasing rate.

D. The object is speeding up at an increasing rate.

E. The object’s speed is changing at a steady rate.



You are given the vx–t graph for an object moving along the x-axis with constant acceleration. Which of the following could you not determine from the information given in this graph alone?

Q2.10

A. the object’s x–acceleration at any time t

B. the object’s x–velocity at any time t

C. the object’s position at any time t

D. more than one of the above

E. misleading question — you could determine all of these from the vx–t graph alone



You are given the vx–t graph for an object moving along the x-axis with constant acceleration. Which of the following could you not determine from the information given in this graph alone?

A2.10

A. the object’s x–acceleration at any time t

B. the object’s x–velocity at any time t

C. the object’s position at any time t

D. more than one of the above

E. misleading question — you could determine all of these from the vx–t graph alone



Q2.11

A. For t > 0, the object is never at rest.

B. The object is at rest at t = 0.5 s.

C. The object is at rest at t = 1.0 s.

D. The object is at rest at t = 2.0 s.

E. More than one of B., C., and D. is correct.

The position of an object moving along the x-axis is given by

x = 5.0 m – (4.0 m/s)t + (2.0 m/s2)t2

Which statement about this object is correct?



A2.11

A. For t > 0, the object is never at rest.

B. The object is at rest at t = 0.5 s.

C. The object is at rest at t = 1.0 s.

D. The object is at rest at t = 2.0 s.

E. More than one of B., C., and D. is correct.

The position of an object moving along the x-axis is given by

x = 5.0 m – (4.0 m/s)t + (2.0 m/s2)t2

Which statement about this object is correct?


Freely Falling Bodies


Freely Falling BodiesThe most familiar example of motion with nearly constant acceleration is a body falling under the influence of the earth’s gravitational attraction.


Earth’s Gravitational Force


• Aristotle thought that heavy bodies fall faster than light bodies.

• Galileo argued that a body should fall with a downward acceleration that is constant and independent of its weight.

Who was right?

Aristotle Ἀριστοτέλης384 – 322 BC

Galileo Galilei1564 – 1642

Freely Falling Bodies


Freely Falling BodiesExperiments show that if the effects of the air can be neglected, Galileo was right; all bodies at a particular location fall with the same downward acceleration, regardless of their size or weight.

If in addition the distance of the fall is small compared with the radius of the earth, and if we ignore the effects due to the earth’s rotation, the acceleration is constant.

This idealized picture is called free fall.


Why is it significant to specify “if the distance of the fall is small compared with the radius of the earth?”


Why is it significant to specify “if the distance of the fall is small compared with the radius of the earth?”


Astronaut Dave Scott tested this himself by dropping a hammer and a feather on the moon.


Gravity is determined by how much mass a given material has, so the more mass an object has, the stronger its gravitational pull. For example, granite is a very dense material with a high level of mass, so it will exert a greater pull than the same volume of a less dense material, such as water. Earth’s mass is distributed between various landforms and features -- such as mountain ranges, oceans, and deep sea trenches -- that all have different mass, which creates an uneven gravity field.

This map, created using data from the Gravity Recovery and Climate Experiment (GRACE) mission, reveals variations in the Earth's gravity field. Dark blue areas show areas with lower than normal gravity, such as the Indian Ocean (far right of image) and the Congo river basin in Africa. Dark red areas indicate areas with higher than normal gravity. The long red bump protruding from the lower left side of the image indicates the Andes Mountains in South America, while the red bump on the upper right side of the image indicates the Himalayan mountains in Asia.




What does “free fall” tell us?



Acceleration is constant and due to gravity




What are the target variables?




What are the target variables?

Position in parts (a) & (c)Velocity in parts (a) & (b)Acceleration in part (d)




Take the origin as the point the ball leaves your hand



Take the positive direction as upward



Take the positive direction as upward






(b) The velocity when the ball is 5 m above the railing?






(c) The maximum height reached and the time that it is reached?








(d) The acceleration of the ball when it is at its maximum height?



Is the acceleration zero at maximum height?




NO, it is still -g




If the acceleration were zero at maximum height what would happen?

NO, it is still -g




If the acceleration were zero at maximum height what would happen?

NO, it is still -g

The ball would stay suspended in mid air!



Can see from the decreasingslope that the ball is slowing down



Can see from the increasingslope that the ball is speeding up



Can see from the increasingslope that the ball is speeding up



Can see from the increasingslope that the ball is speeding up Position has changed

Velocity has changedAcceleration is constant


Find when the ball is 5 m below the roof railing


Find when the ball is 5 m below the roof railing


Even the smoothest sports car does not move with constant acceleration.

The motion may be integrated over many small time windows.







Relative Velocity


Relative Velocity


Relative Velocity

A conductor and a passenger on a moving train may both be moving at a rapid clip toward their destination, yet, the conductor can still move toward your seat to check your ticket.


Relative Velocity



Relative Velocity



Summary


Summary


Summary


Summary


Summary


Documents

Motion Along a Straight Line - University of Texas at Dallasdavid.lary/CM/... · 8/31/2011 · Motion Along a Straight Line Wednesday, August 31, 11. Kinematics (from Greek κινεῖν,