How Can You Calculate Total Distance Traveled Without Integration?

[Mola]

Homework Statement

A roller coaster ride lasts only 60 seconds. What is the total distance traveled during the 60 seconds, given the position of the roller as a function of time is s(t) = 4cos(t)sin(t).

Homework Equations

Position as a function of time S(t) = 4cos(t)sin(t)

The Attempt at a Solution

I solved for velocity by differentiating and i got V(t) = 4[cos²(t) - sin²(t)].

I was going to find the absolute integral of the velocity to get the total distance, but the teacher said we cannot use integration because we did not reach that section yet. How else can I find the total distance without integration?

 

[pat666]

stick t=60 into the equation, there is no need or value in integrating S(t) its already in the form you need,,

 

[rock.freak667]

Since you differentiated it, and were going to integrate it, wouldn't you get back the function s(t)?

So you only really need to get s(60)

 

[pat666]

Just fyi your differentiation isn't right. should be 8cos^2(t)-4

 

[rock.freak667]

Just fyi your differentiation isn't right. should be 8cos^2(t)-4

s=4sintcost =2 sin2t

ds/dt = 4cos2t = 4[cos²t-sin²t]

 

[pat666]

sorry just realized using trig identities that exactly the same thing... should check before saying you were wrong
haven't differentiated by hand in years ti-89...

 

[Mola]

Well I was confused because I thought S(60) -S(0) is displacement instead of total distance traveled??

I used S(60) -S(0) and got 1.16(thinking that is the displacement). Divided that by time( = 60s) and got average velocity of .019m/s.

 

[pat666]

why do you want a velocity??/

 

[Mola]

Because I'm trying to compare 3 different roller coasters to see which one covers the most distance, the greatest velocity and the biggest acceleration during the 60 second ride.
The equation S(t) = 4sin(t)cos(t) is the POSITION of the roller coaster with respect to time for one of the roller coasters.

 

[pat666]

If you are to get technical displacement is distance when you chose your reference point at the start which is the only thing you can do (assuming straight motion and no reverse)! Why are you trying to be so technical it is an extremely easy question that you are over complicating. the distance traveled for this roller coaster is s(60) which is exactly sqrt(3) or 1.7 metres. put up the other part of the question with acceleration and velocity so we can see the question from your context.

 

[Mola]

A ride on each of 3 different roller coasters lasts exactly 60 seconds. The horizontal position of each of the roller coasters (in meters) as a function of time
(in seconds) is given by

Streaker: S(t) = 4sin(t)cos(t)

Redhawk: S(t) = e^t/(t²-400)

Dragster: S(t) = (3600 - t²)^3/2

Find the the fastest coaster that covers the largest total distance. Rank
the 3 coasters in terms of 1) velocity, 2) acceleration and 3) total distance traveled.

 

[pat666]

Ok now i see what you were trying to do.

Streaker- s=sqrt(3) , v=sqrt(3)/60=0.0288 , a=v2-v1/t=sqrt(3)/120=0.0144m/s^2
Redhawk s=3.57*10^22 m, v=5.94*10^20m/s, a=9.9*10^14m/s^2
Dragster s=0,v=0,a=0

this is obviously not motion in a straight line, this question is impossible without the use of integration and differentiation. I see why you were having so much trouble with it, is the chapter it is in concerned with integration?

 

[Mola]

No, we haven't reached integration yet so he is not expecting us to solve it by integration. We're just about to finish differentiation.
Thank you so much for the help. I appreciate it. It was supposed to be challenging he said, so I was thinking it's not going to be that simple.

But how about if these roller coasters were to go back and forth(forward and reverse) during the 60 second period? Do you think it will be solvable without integration?

 

[pat666]

what you are talking about is periodic motion. but i take it that your teacher is fine with you using differentiation which i now realize is all you need. differentiate s to get v and the double differentiate s to get a.