Question

A car is traveling at 88ft/s when the driver applies the brakes to avoid hitting a child. After t seconds, the car is s(t)=88t-8t^2 feet from the point where the brakes were first applied. How long does it take for the car to come to a stop, and how far does it travel before stopping?

Answer 1

Hint: differentiate this ^_^

Answer 2

I'll tell you the next step after you get this differentiated.

Answer 3

by differentiate, u mean derive, right?

Answer 4

Well, derive actually means something different, but I know what you mean, so yes. ^_^

Answer 5

I'm assuming you mean you want to do d/dx, which is correct.

Answer 6

yes :)

Answer 7

s'(t)=88-16t

Answer 8

Perfect!

Answer 9

Now, I'm also in calculus 1 right now, so I wanna explain this part really carefully as it's VERY important:

Answer 10

almost any time you see feet per second or inches per degree, or anything per anything, it's almost always a derivative.

Answer 11

Like... if you had miles per hour, it'd look like dm/dh
or m'(h). Does that make sense so far?

Answer 12

yes, it does.

Answer 13

Awesome! So can you tell me what derivative is given to us in the original problem?

Answer 14

Repost: A car is traveling at 88ft/s when the driver applies the brakes to avoid hitting a child. After t seconds, the car is s(t)=88t-8t^2 feet from the point where the brakes were first applied. How long does it take for the car to come to a stop, and how far does it travel before stopping?

Answer 15

Here's a little hint from physics class: s usually means distance.

Answer 16

the 88ft/s???

Answer 17

Bingo!

Answer 18

So in derivative form, how would we write that?

Answer 19

ft'(s)=88 ???

Answer 20

very close.
It's actually s't
s't is the same as
ds/dt meaning "for every t [which we know to mean seconds], the car moves s [which we know to mean feet]"

Answer 21

ok,and then?

Answer 22

Bleh, sorry... I had apparently mistaken how to do this.

After you differentiated to
s'(t)=88-16t

divide the 88 by 16 to get 5.5 seconds.

You can plug this in the original equation to get
242 feet. However...... double check this with someone else. I'm not feeling as comfortable with this anymore. :/

Especially since the first time I'd done this question a few months ago, I recall there was a mention of a kid that was about to be hit as well as some distance.

Answer 23

The numbers do match closely with real life example of cars moving at 60mph (which is 88 ft/s).

Answer 24

"60 MPH = 88 fps. (fps=1.467 * MPH). If the vehicle deceleration rate is 20 fpsps (rather than the previously calculated 15 fps), then stopping time = 88/20 = 4.4 seconds. Since there is a 1 second delay (driver reaction time) in hitting your brakes (both recognition and reaction time is often 2 seconds), the total time to stop is 5.4 seconds to 6.4 seconds.

To determine how far the vehicle will travel while braking, use the formula of 1/2 the initial velocity multiplied by the time required to stop. In this case, this works out to be .5 * 88 * 4.4 = 193.6 feet, plus a reaction time of either 88 feet for a second delay in reaction time, or 176 feet for two seconds reaction time. That yields 281.6 feet or 369.6 when added to the base stopping distance of 193.6 feet. If the driver is very responsive and takes only a half a second to react, the distance is reduced to 237.6 feet. Notice that the reaction time is a huge factor since it is at initial velocity."

So the answer's most likely correct, but I still am apprehensive, especially since I pretty much made up the rule to divide 88 by 16. It doesn't follow in line with any other calculus that I've learned.