Constant Acceleration With Two Objects

[eltavo809]

Homework Statement

A train pulls away from a station with a constant
acceleration of 0.40 m/s2. A passenger arrives at the
track 6.0 s after the end of the train has passed the
very same point. What is the slowest constant speed at
which she can run and catch the train?

Homework Equations

X = X₀ + V₀ * t + 0.5 * a * t²
V = V₀ + a * t

The Attempt at a Solution

The passenger arrieves 6 sec. latter, so: t_p = t_t - 6sec

Passenger

X = V₀ * ( t - 6 sec)

Train

X = 0.5 * 0.4 m/s² * t²

Both, train and passenger, have to be at the same place at the same time in order to achieve an encounter, so:

x_train = x _passenger

V₀ * ( t - 6 sec) = 0.2 m/s² * t²So here is the problem. I know I have to find out V₀ , but I have 3 variables and 2 equations. Any ideas? This is supose to be easy... :(

 

[dontdisturbmycircles]

For the passenger, X is Vo*(t-6) not +, careful!

An easy way to check this is plug in t=6s, he should be at x=0.

Then your strategy of setting X1=X2 is correct.

You should first get a function of time as a function of Vo. Then try to remember how you can find the minimum value of a function!

 

[Delphi51]

After the 6 s head start, the train is moving at 2.4 m/s and has gone 7.2 m.
Easier to let t be the time after this point.

(edited - I had a mistake earlier)

 

[dontdisturbmycircles]

We know the position of the person is X1 = V0 * ( t - 6 )

The position of the train is X2 = 0.2 * t^2

All he needs to do is set X1=X2 and find a function of the time taken to catch up in terms of Vo, he can then use calculus to determine the best choice of Vo :)

 

[Delphi51]

Okay, I see it - had confused distance and velocity and got a v function with no minimum.
How interesting, at speed 5 the person catches the train in 9 seconds. At speed 4, she never catches it.

 

[eltavo809]

So, after some basic algebra, I got this:

v₀ * (t - 6) = 0.2 * t²

-0.2 t² + v₀ t - v₀ 6 = 0

From the quadratic formula, I get:

(- v₀ t +/- √ v₀² + 4 * 0.2 * 6 v₀) / 0.4

This should give me the time at which she catches the train, right?

 

[dontdisturbmycircles]

Using the quadratic formula would be meaningless here, you don't want the values of Vo such that t=0 (which is what the quadratic formula would give you)!

Do you know how to take the derivative of a function and find its minimum value? :) If not I can just show you a way to find it =-).

 

[eltavo809]

Hey, dontdisturbmycircles, first of all let me thank you all your time and dedication.

Yes, i know how to derivate and find a minimun of a function. I just was not expecting to come to that kind of solution. Having said that, thanks again for the help. I got it now!

 

[dontdisturbmycircles]

You are welcome ;).