Relative Velocity problem - stuck, lacking ideas.

[Siracuse]

Homework Statement

Halliday/Resnick/Walker, 1 - Mechanics, 7th Edition.
Chap. 4, Exercise 57 - Relative Movement in One Dimension

(I'm sorry if translation sounds confusing at first, the book is in Portuguese)

A suspicious man runs as fast as he can on a conveyor belt, taking aprox. 2.5 seconds to reach the other side. Then, a guard(lit. "security agent") shows up and the man tries to run back as fast as he can to the start point, taking 10 seconds. What is the (lit.) "reason" between the man's speed and the conveyor belt's speed?

Homework Equations

The "reason" (don't know the right word) that the problem is asking for is basically:

(X_m)/(X_b), where v_m is the man's speed and v_b is the conveyor belt's speed.

Revelant equation for R. M. in one Dimension:

X_pa = X_pb + X_ba; for velocity and acceleration, just a matter of taking the derivatives.

The Attempt at a Solution

All I could do was a superficial analysis of the problem: if it took less time to walk from x_0(start point) to x_f(end point) while on the conveyor belt when compared to the second displacement, obviously the vectors v_m and v_b are in the same direction (positive in respect to the x-axis) in the first displacement.

I have 2 known referentials: the man and the conveyor belt. A third one, the "stationary" referential, would be the floor (f). So, renaming the variables the eq. for R. M.,

X_mf = X_mb + X_bf

From here...no idea what to do. How exactly do I relate the man's velocity on the conveyor belt and the man's velocity while running by foot?

 

[mgb_phys]

I think you are over complicating it. Reason = relation / ratio.

The man runs a distance x in 2.5 seconds with the belt and 10secs against it.
so he is doing x/2.5 m/s in one case and x/10 m/s in the other.
We assume his legs are running at the same speed in both cases.
We also know he must be running faster than belt or he would never get back ( this let's us check we have the signs correct!)
Let v=speed of man u=speed of belt

First case u+v = x/2.5 second case u-v=x/10
X is the same, so x = 2.5(u + v) = 10(u-v)
2.5v+10v = 10u-2.5u so 12.5u = 7.5v then u = 7.5/12.5 v or v = 12.5/7.5 u

 

[Siracuse]

I think you are over complicating it. Reason = relation / ratio.

I do that very often.

The man runs a distance x in 2.5 seconds with the belt and 10secs against it.
so he is doing x/2.5 m/s in one case and x/10 m/s in the other.
We assume his legs are running at the same speed in both cases.
We also know he must be running faster than belt or he would never get back ( this let's us check we have the signs correct!)
Let v=speed of man u=speed of belt

First case u+v = x/2.5 second case u-v=x/10
X is the same, so x = 2.5(u + v) = 10(u-v)
2.5v+10v = 10u-2.5u so 12.5u = 7.5v then u = 7.5/12.5 v or v = 12.5/7.5 u

Sounds logically correct and more precise than my thinking. The answer's correct.

Nevermind that, got the concept...well, thanks nonetheless. :)