1. How far will an object move in one second if its average speed is 5 m/s?
A. 5 meters
2. How far will a freely falling object have fallen from a position of rest when its instantaneous speed is 10 m/s?
A. 75 meters
3. An object dropped from rest and falls freely. After 6 seconds, calculate its instantaneous speed, average speed, and distance fallen.
A. Instantaneous speed: 58.8 m/s
Average Speed : 53.9 m/s
Distance : 313.6 meters
4. If a freely falling rock were equipped with an odometer, would the readings for the distance fallen each second stay the same...

- calculusxy

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- calculusxy

...increase with tie, or decrease with time?
A. The instantaneous speed would increase so the distance would increase as well.

- calculusxy

- calculusxy

Can you make sure the answers?

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## More answers

- ybarrap

A. Is correct because
$$
5 \frac{m}{\cancel{s}}\times 1~\cancel{s}=5~m
$$
Notice how the seconds cancel each other out and you are left with just meters ,\(m\)?

- calculusxy

Yeah

- ybarrap

For 2 you need to use the following equation
$$
v^2 = u^2 + 2as
$$
where u is the initial velocity, a is the acceleration of gravity, s is the distance traveled and v is the final velocity: https://en.wikipedia.org/wiki/Equations_of_motion#Kinematic_equations_for_one_particle
So
$$
10^2=0^2+2s\times 9.81\\
\implies s=\frac{100}{2\times 9.81}=5~m
$$
Does this make sense?

- calculusxy

Answer would be 5M?

- ybarrap

Yes

- calculusxy

how would it be 5m if it is travelling 10m/s

- ybarrap

|dw:1436316669347:dw|
(ignore the t=1/2 seconds above)
It takes
$$
v/a=10/9.81=1.01\text{ second}
$$
to go from 0 to 10 m/s
In that time period, it is accelerating. If it were going a constant speed of 10 m/s then is 1 second it would have traveled 10 m, but it didn't START at 10 m/s, it started at 0 m/s.
Right?

- calculusxy

yes

- ybarrap

For # 3
Instantaneous speed:
$$
v=at\\
v=9.8\times 6=58.8~m/s
$$
Average speed:
$$
\frac{1}{6}\int_0^6 at~dt=\frac{1}{6}a\frac{t^2}{2}|_0^6=\frac{9.8\times 6^2}{12}=29.4~m/s
$$
distance
$$
s=\frac{1}{2}at^2=\frac{1}{2}9.8(6)^2=176.4~m
$$
Does this make sense?

- calculusxy

Sorry but I am just a rising eighth grader so i don't understand the average speed part.

- ybarrap

How did you determine average?

- ybarrap

You can graph it, that's another way, want to try that?

- calculusxy

what i did was:
5 sec = 49 m/s
6 sec = 58.8 m/s
49 + 58.8 / 2 = 53.9

- ybarrap

Why did you use 5 seconds here?

- calculusxy

well that's what my teacher said for me to use

- calculusxy

2 seconds => average speed
19.6 + 9.8 /2 = 14.7 m/s

- calculusxy

9.8 (1st second)
19.6 (2nd second)

- ybarrap

Yes, that is a good approach, keep going

- calculusxy

so 53.9 is correct?

- ybarrap

It will be an estimate... but if you want a formula, use:
$$
\frac{v_{final}-v_{initial}}{2}=\frac{v_{final}-0}{2}=\frac{a t}{2}
$$
Because the initial velocity is 0 and final velocity = \(at\).

- calculusxy

a = ?

- ybarrap

Average speed should be 9.8*6/2.
a is the acceleration of gravity = 9.8 m/s^2

- calculusxy

29.4

- ybarrap

yes

- calculusxy

okay

- calculusxy

next

- ybarrap

So did you get the distance traveled?

- calculusxy

for #4?

- ybarrap

#3

- calculusxy

yeah

- ybarrap

176.4 m ?

- calculusxy

yes

- ybarrap

Ok for the final one...

- ybarrap

An odometers measures distance traveled, right?

- calculusxy

yes

- ybarrap

And acceleration is change in velocity per unit time, so your speed is changing every second, in fact its increasing every second, right?

- calculusxy

yes

- ybarrap

Ok, so velocity is changing. If in one second you are going 1 m/s and in the next second you are going 2 m/s then the distance you are traveling every second is increasing. So the odometer will show for the 1st second one meter and the odometer will show for the second second, 2 meters. Do you see that?

- calculusxy

yes so it's increasing right?

- ybarrap

yes, because velocity is increasing. If you were slowing down, then the odometer would do the opposite

- calculusxy

thank you so much!

- ybarrap

you're welcome!!

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