Chapter 2 : 1-D Kinematics:Velocity & acceleration.

brontidegrrrMechanics

Nov 14, 2013 (3 years and 6 months ago)

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Chapter 2 : 1
-
D Kinematics:Velocity & acceleration.

Updated 9/6/11


Try to do as many prob
l
ems as you can, emphasizing the ones we do in class. There always will be
more problems than many of you can do, and there will be some very challenging problems to
ward the end
of the set. Do your best, ask questions,
use the problems to learn
!!

1. a) Clearly explain the difference between
position

and
displacement
, giving examples. Explain why you
need to read carefully when the term "distance" appears in a problem.



b) Clearly distinguish between the concepts of
instantaneous

and
average

velocity giving examples.
During what type of motion are these two the same?



c) Clearly explain the difference between
instantaneous

velocity,
change

in velocity and
accelera
tion
.
Give examples that illustrate the difference.



d) Distinguish between
constant velocity
,
constant acceleration
, and
non
-
constant acceleration
. Illustrate
with examples.


e) Distinguish between +
acceleration

and

acceleration
. Also distinguish b
etween “acceleration” and
“deceleration”.

2. In preparation for the “Graphing Constant Acceleration” lab, sketch
rough

graphs of position, velocity,
and acceleration vs. time for the following cases of
constant
acceleration:



a) a=0, v>0; and also a=0,

v<0;

b) a>0, v
o
>0; and also a>0, v
o
<0;

c) a<0, v
o
>0; and also a<0, v
o
<0


Answer these questions about the graphs you sketched:



d) In which of the cases above was the
speed increasing
?...
speed decreasing
?



e) In which of the cases above can the ob
ject turn around?



f) Is it possible for the speed to be
zero

and the acceleration to be
non
-
zero

at any time? Explain.

3. The following chart presents a number of motion cases (a through f) for which either a
verbal

description
or a
graphical

descripti
on (either position, or velocity, or acceleration vs. time) is given. Fill in the
missing

graphs or verbal description for each case. Ignore any points of
abrupt change
. Assume that any initial
position or velocity not given is zero.


a)

b)

c)

d)

e)

f)

De
scription


A car is moving
with constant
velocity, then it
accelerates to a
higher constant
velocity.


A mass
hanging from
a spring is
moving up
and down.


A ball is
tossed
upwards, it
rises in the air
and returns to
the ground.

x vs. t








v vs. t









a vs. t









4. A
motion diagram

is an illustration showing the
velocity and acceleration vectors

of the moving object.
Draw
motion diagrams

for cases (b),


(c), & (e) in the problem above.


a) Draw enough
instantaneous velocity

vectors a
t equal time intervals to clearly illustrate each case.


b) Draw
change in velocity

vectors for each time interval and then draw
acceleration

vectors for each time
interval. Ignore the points of abrupt change.

5. An
equation of motion

describes algebra
ically how the motion of an object depends on time. Write
equations of motion

describing cases (d), (f), & (e) above [case (e) required three sets of equations], that is:


a) Write the equation of position vs. time [x(t)] for each case.


b) Write t
he equation of velocity vs. time [v(t)] for each case.


c) Write the equation of acceleration vs. time [a(t)] for each case.

6. When an object changes its motion abruptly in a short amount of time (as we have seen in many
examples) we ignore the s
hort time intervals for the sake of
simplicity

and because they are
negligible

compared to the longer intervals of motion. These abrupt changes often are drawn as a sharp angle or broken
line in the motion graphs. Explain why such abrupt changes are imposs
ible in the real world.

7. There are two formulas that we can use to determine average velocity: v
ave
=

x/

t or v
ave
=(v
i
+ v
f
)/2.



a) Which one of these formulas is
always

correct (by definition) and which formula is only correct under
certain condition
s? What are the limitations of the “other” formula.



b) A car travels 12 miles using two different constant speeds. The car travels
half the distance

with a
speed of 10 mph and the other half distance with a speed of 30 mph. Determine the average speed.



c) The car travels the same 12 miles back but this time the driver spends
half the total time

of the trip
traveling at 10 mph and the other half time traveling at 30 mph. What’s the average speed for the return trip?



d) Sketch position vs. time gr
aphs for both trips. Which trip took longer (b) or (c)?



e) If you changed your speed gradually from 10 mph to 30 mph during the 12 miles trip, which of the two
cases above would have the same average velocity?



f) Prove (or give a logical argument)
that the “half
-
time” average velocity will always be faster than the
“half
-
distance” average velocity.

8. A car (A) is waiting for the red light to change at an intersection. When the light changes the car
accelerates at a rate of 1 m/s
2
. At the moment th
at car begins to move a second car (B) passes it going at a
constant speed of 8 m/s.



a) If neither car changes its motion, how soon will it take for car A to match the speed of car B? Which
car is ahead at that time?




b) If neither car changes its
motion, how soon will it take for car A to overtake car B? Which car is
moving faster at that time? How far from the entrance of the intersection does this happen?




c) Draw graphs of position and velocity vs. time for the two cars. Use a single positio
n graphs and a single
velocity graphs for both cars. Make sure your graphs agree with your answers to (a) and (b).



d) If car A stops accelerating after 12 sec, when will it overtake B? At what location will this happen?

9. A man is running to catch a b
us at his top speed of 6 m/s. The man is 25 m behind the bus when the bus
starts to pull away from the curve with an acceleration of 1 m/s
2
.




a) Will the man catch the bus? If
yes
, find out when and where it happens. If
no
, determine how close he
got t
o the bus and the minimum speed he needed catch it.



b) Draw a position graph that includes both the motion of the man and the bus in the same scale.



c) Repeat the problem assuming that the bus is accelerating at a rate of 0.5 m/s
2
.

10. Think about
what happens when you toss a ball up in the air and it rises high above your head.



a) Roughly
estimate

the average acceleration of the ball
while in the hand

and compare to its acceleration
when it’s

in the air
. Which acceleration is higher in magnitu
de? Do they have the same directions?
Hint
:
Compare how far the ball travels in your hand to how far it travels in the air and recall that g~10 m/s
2
.



b) Now assume the ball moves 1 meter upward while in the hand, and that it is 2 m above the ground
when it leaves your hand. In addition assume that the ball is moving with an initial speed upwards of 12 m/s
as it leaves your hand. Draw graphs of position, velocity, and acceleration vs. time for the motion of the ball
from the time it’s in the hand to w
hen it hits the ground. Assume air resistance is negligible.



c) How long is the ball in contact with the hand? How long is the ball in free
-
fall?



d) How fast is the ball moving when it impacts the ground?



e) Repeat the problem reversing the di
rection of + and
-

in your solution. Verify that this does not change
the value of the answers.

11. We often ignore air resistance for the sake of simplicity. The air produces an acceleration that opposes
the motion and, that in addition, depends on the sp
eed of the object itself (the greater the speed the greater
the effect of air resistance). Consider the motion of the ball in free
-
fall in the problem above, but now let’s
consider the effect of air resistance in a
qualitative

way.



a) Show how the shap
e of the graphs of motion in the problem above would change as a result of air
resistance. Do not worry about numerical values; concentrate on the overall shape of the graphs. Explain the
changes you made.



b) How does the time for the ball to go up a g
iven distance compare to its time coming down for
the same
distance

in the case of non
-
negligible air resistance? Justify your answer.

12. A balloon is rising with a constant speed of 5 m/s. It carries a basket with a person inside. When the
balloon is a 1
20 m above the

ground the person holds an object outside the basket and lets it go.



a) Describe the motion of the object as seen by a person on the ground and as seen by the person in the
balloon’s basket.



b) What maximum height does the object rise
above the ground?



c) How long does it take the object to reach its maximum height? ...to hit the ground?


d) How fast is it going when it hits the ground?





e) If the person had thrown the object down ward with a speed of 10 m/s
relative

to the b
alloon (instead of
merely letting go), how long would the object take to hit the ground in that case?

13. A ball is thrown straight upwards with a velocity v
o
. At the same time a ball is
dropped

from a height H
above the ground. Determine answers in terms
of H, g, and v
o
.



a) At what time do the balls cross paths?



b) Suppose v
o
= 5 m/s and H= 6 m. Where would the balls cross paths?



c) In terms of v
o

and g, derive an expression for the maximum possible value of H that will allow the balls
to cross
paths
above the ground
.

14. A man is trying to lift a weight using various rope and pulley set
-
ups as illustrated below. Assume that
the man is pulling his end of the rope with a speed
v
o
.
This problem is not so important in this chapter but
will become

more important in future chapters.



a) For each set
-
up determine the speed at which the weight rises. Explain your reasoning clearly.



b) If the man is accelerating his end of the rope at a rate
"a"
, at what rate is the weight accelerating in each
c
ase.

(i)





(ii)





(iii)



15. An elevator has a height of 3 m. While the elevator is moving a bolt inside the elevator drops from the
ceiling to the floor. Determine the time it takes for the bolt to hit the elevator floor in the following cases.



a
) The elevator is moving up with a constant speed of 1 m/s.



b) The elevator is accelerating down at a rate of 2 m/s
2

and the speed of the elevator is up 1 m/s at the
instant the bolt comes loose.



c) Describe the motion of the bolt in (b) as seen by

an observer on the ground....as seen by an observer
inside the elevator.



d) Solve problem (b) using the
frame of reference of the observer inside the elevator

and show that the
answer is the same.

v
o

?

v
o

?

v
o

?



e) Repeat the problem assuming that the elevator
is accelerating up at a rate of 2 m/s
2

and the speed of the
elevator is up 1 m/s at the instant the bolt comes loose.

16. The breaks are applied to a car traveling with speed v
o

and it comes to a stop in a distance "d". We
assume that the
acceleration due
to the breaks is constant

for this car regardless of speed.



a) If the initial speed had been 2v
o
, how much longer would it take the car to stop?



b) In driving school they
suggest

that you allow one car length of stopping distance for every 10 mph o
f
speed of your car. Is this consistent with a constant deceleration from your brakes? Explain.



c) In reality one has to allow for the "reaction time" that it takes before the foot hits the brakes. During
this time the car moves through a "reaction dis
tance" at the initial constant velocity and this adds to the
"braking distance" to make up the total "stopping distance". The following table gives typical values:

Initial speed (m/s)



Reaction distance


(m)


Braking distance (m)

Stopping distance (m)

10

7.5

5.0

12.5

20

15

20

35

30

22.5

45

67.5



What is the "reaction time" implied by the data in the chart above? What is the acceleration generated by
the brakes? Is the acceleration constant as we assumed earlier?



d) Determine the car’s stopping d
istance if the initial speed of the driver is 25 m/s?

17. A block slides over a rough patch of surface 40 cm long. When it enters the patch the speed of block is
12 cm/s and when it emerges its speed is 6 cm/s.



a) Determine the time the block spends m
oving over the rough patch.



b) Determine the acceleration of the block.



c) With the same acceleration, how much larger should the patch be in order to stop the block altogether?



d) A second block enters the patch 1 s later moving with velocity
15 cm/s, will it overtake the first block
within the patch? Assume both blocks have the same acceleration moving over the rough patch.

18. A train is traveling from station A to station B a distance D away. Starting at A the train first accelerates
uniform
ly to a maximum speed v
max
then it decelerates uniformly to come to a stop at B. The train spends
twice as much time decelerating as it spends accelerating.



a) Draw graph of position, velocity and acceleration vs. time for the entire trip.



b) Deter
mine the average speed during acceleration, ...during deceleration,...and overall.



c) Determine the time for the complete trip in terms of v
max

and D.



d) Determine the accelerations.



e) Would it have changed the overall time for the trip if the

train had accelerated more quickly initially to
the maximum speed and then taken longer to some to a stop? Justify your answer.

Problems below require calculus to solve:


19. Review the proof of the kinematics formula for position under a constant acceler
ation [x = at
2
/2 + v
o
t +
x
o
] which was done in class. Then proceed to proof the kinematics formula without time [2a

x = v
2
-
v
o
2
].
Hint
: Rewrite the definition of acceleration with the help of the chain rule as: a=dv/dt= (dv/dx)(dx/dt)]. Set
up the appropriate integral and solve.

20. Air resistance always opposes the motion and depends on the velocity of the object. It a
lso depends on
some of the physical properties of the air (like its density) and of the object (such as its shape). If all the
physical properties that affect air resistance are combined into a single constant "C", the acceleration due to
air resistance ca
n be written as
a
=
-
Cv
2
(
i
) (where
i
here indicates the direction of the velocity).



a) Explain the meaning of the negative sign in this formula.



b) Derive a function of the velocity as a function of time due to air resistance on an object with init
ial
speed v
o
. Graph this function. Compare to the example of “fluid resistance” done in class.


c) How long would it take the object to slow down to half its initial velocity?…to come to rest?



d) The acceleration due to gravity can make an object "t
urn around" in free
-
fall. Can air resistance alone
make an object "turn around? Justify your answer.

21. Galileo, who gave us our basic definition of acceleration, noticed that objects moving under the effect of
gravity traveled "odd multiples of distance

in successive time intervals". This means that if an object falls a
distance "x" in a time "t" from rest , it would fall a distance "3x" during the next “t” interval and "5x" the
time interval after that…etc.



a) Show that this is consistent with our k
inematics formula x=gt
2
/2 + v
o
t.



b) Show that this is consistent with the meaning of a constant acceleration.



c) Galileo had considered defining acceleration as a=dv/dx instead of dv/dt. Derive a kinematics formula
for position vs. time for the cas
e in which “a=dv/dx” is a constant.