## calculusxy one year ago 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...

1. calculusxy

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

2. calculusxy

@jim_thompson5910

3. calculusxy

Can you make sure the answers?

4. 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 ,$$m$$?

5. calculusxy

Yeah

6. 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?

7. calculusxy

8. ybarrap

Yes

9. calculusxy

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

10. 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?

11. calculusxy

yes

12. 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?

13. calculusxy

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

14. ybarrap

How did you determine average?

15. ybarrap

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

16. calculusxy

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

17. ybarrap

Why did you use 5 seconds here?

18. calculusxy

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

19. calculusxy

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

20. calculusxy

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

21. ybarrap

Yes, that is a good approach, keep going

22. calculusxy

so 53.9 is correct?

23. 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 = $$at$$.

24. calculusxy

a = ?

25. ybarrap

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

26. calculusxy

29.4

27. ybarrap

yes

28. calculusxy

okay

29. calculusxy

next

30. ybarrap

So did you get the distance traveled?

31. calculusxy

for #4?

32. ybarrap

#3

33. calculusxy

yeah

34. ybarrap

176.4 m ?

35. calculusxy

yes

36. ybarrap

Ok for the final one...

37. ybarrap

An odometers measures distance traveled, right?

38. calculusxy

yes

39. ybarrap

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

40. calculusxy

yes

41. 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?

42. calculusxy

yes so it's increasing right?

43. ybarrap

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

44. calculusxy

thank you so much!

45. ybarrap

you're welcome!!