Is the runner running continuously?

[Mooch22]

I have no idea of where to start this problem! Some help, please?

The rate of a runner running the 400 meter run in 50 seconds is given by a differentiable function r(t). The table below shows the rate as measured every 5 seconds:

Time (seconds) Rate (m/sec)
0 :arrow: 0
5 :arrow: 5
10 :arrow: 7
15 :arrow: 10
20 :arrow: 12
25 :arrow: 13
30 :arrow: 11
35 :arrow: 9
40 :arrow: 8
45 :arrow: 7
50 :arrow: 6

A.) Explain why r(t) is a continuous function.

B.) Given some time (t=c) on (0, 50) and using correct units, explain the meaning of:

lim ((r(c)-r(t))/(c-t))
t -> c

C.) Is there some time on the interval (0, 50) such that r'(t)=0? Explain.

D.) Find the average acceleration on [10, 40].

E.) Approximate as accurately as possible the acceleration of the runner at t=30 seconds algebraically

 

[tkhunny]

a) Thought question. How does anything change speeds? If you watch fine enough increments, it will look continuous. In this case, we're talking about a running human. It won't take very fine increments unless he smacks into a wall, or something.

b) Instantaneous rate of acceleration - m/sec²

c) That would be the same question as asking if the speed fails to change from one moment to the next. That SEEMS unlikely. The data SHOWS no such event, but I think the data intervals are much to broad to make any sweeping assumptions about NEVER occuring.

Are you started, yet?

 

[Mooch22]

HELP.... PLEASE!!! I HAVE TO HAVE THIS FINISHED TOMORROW!!!

I got part A... but what in the world do you do with b and c?!?!?

Help please!

 

[Gene]

But TKH gave you the answer to b). Instanious acceleration.

c) TKH missed here. Since r'(t) is the acceleration and around 25 seconds he stopped accelerating (positive dv/dt) and started decelerating (negative dv/dt) there must be a r'(t) = dv/dt = 0 in there somewhere.

 

[Daniel_Feldman]

Isn't part c) Rolle's Theorem?

 

[Mooch22]

REPLY....

I hate to sound dumb on this one...but what is instantaneous acceleration? I'm in such desperate need for this by tomorrow morning... please please help... the only part i understand is a!

 

[Gene]

That is the acceleration at a certain very small fraction of a second (c-t->0) or at an instant.
He has different speeds at different times so he must be accelerating or deccelerating to cause the speed change. dv=a*dt. If dv is negative a is negative.

 

[Mooch22]

Ok--I think I get b, but... (please understand that I am trying very hard...my teacher doesn't like to help all that much outside of class, so this site is my saving grace!!)

...so how do i find r'(t) if I do not have a function? do i punch in the values in the calculator and do a best fit line? and then take the derivative of that function?

...and for the average acceleration... do i average the numbers in the given information?

You don't know how much I appreciate your help!

 

[Gene]

You don't have to find it, just prove it exists. If you were on the third floor of a building this morning and are in the basement now don't you suspect the police would know you were on the first floor sometime today? They don't have to know what you did in between. Same with the (continuous) accelerations. Some are positive (third floor) and some are negative (the basement. Between them there must be a zero (first floor.)