calculusxy
  • calculusxy
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...
Mathematics
chestercat
  • chestercat
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calculusxy
  • calculusxy
...increase with tie, or decrease with time? A. The instantaneous speed would increase so the distance would increase as well.
calculusxy
  • calculusxy
calculusxy
  • calculusxy
Can you make sure the answers?

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ybarrap
  • 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
  • calculusxy
Yeah
ybarrap
  • 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
  • calculusxy
Answer would be 5M?
ybarrap
  • ybarrap
Yes
calculusxy
  • calculusxy
how would it be 5m if it is travelling 10m/s
ybarrap
  • 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
  • calculusxy
yes
ybarrap
  • 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
  • calculusxy
Sorry but I am just a rising eighth grader so i don't understand the average speed part.
ybarrap
  • ybarrap
How did you determine average?
ybarrap
  • ybarrap
You can graph it, that's another way, want to try that?
calculusxy
  • calculusxy
what i did was: 5 sec = 49 m/s 6 sec = 58.8 m/s 49 + 58.8 / 2 = 53.9
ybarrap
  • ybarrap
Why did you use 5 seconds here?
calculusxy
  • calculusxy
well that's what my teacher said for me to use
calculusxy
  • calculusxy
2 seconds => average speed 19.6 + 9.8 /2 = 14.7 m/s
calculusxy
  • calculusxy
9.8 (1st second) 19.6 (2nd second)
ybarrap
  • ybarrap
Yes, that is a good approach, keep going
calculusxy
  • calculusxy
so 53.9 is correct?
ybarrap
  • 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
  • calculusxy
a = ?
ybarrap
  • ybarrap
Average speed should be 9.8*6/2. a is the acceleration of gravity = 9.8 m/s^2
calculusxy
  • calculusxy
29.4
ybarrap
  • ybarrap
yes
calculusxy
  • calculusxy
okay
calculusxy
  • calculusxy
next
ybarrap
  • ybarrap
So did you get the distance traveled?
calculusxy
  • calculusxy
for #4?
ybarrap
  • ybarrap
#3
calculusxy
  • calculusxy
yeah
ybarrap
  • ybarrap
176.4 m ?
calculusxy
  • calculusxy
yes
ybarrap
  • ybarrap
Ok for the final one...
ybarrap
  • ybarrap
An odometers measures distance traveled, right?
calculusxy
  • calculusxy
yes
ybarrap
  • 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
  • calculusxy
yes
ybarrap
  • 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
  • calculusxy
yes so it's increasing right?
ybarrap
  • ybarrap
yes, because velocity is increasing. If you were slowing down, then the odometer would do the opposite
calculusxy
  • calculusxy
thank you so much!
ybarrap
  • ybarrap
you're welcome!!

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