how fast can Xander run?

Question

Xander was walking across a train bridge and was 2/3 rds of the way across when he saw a train coming at 45 mph toward him.  Xander is fast enough that if he ran the rest of the way he'd make it. If he ran back he'd just make it. How fast can Xander run?

Answer 1

Miki,

 

Let's let the length of the bridge = L, then Xander needs to cross a distance L/3 to reach the end of the track.

At the same time let the train be a distance X away from the end of the track.  The time t₁ for Xander and the train to reach the end of the track is the same, so with V_{x ⁼} Xander's speed and V_t = the trains speed since distance = velocity x time:

 

(1) L/3 = V_x ·t1  and

(2) X = V_t · t₁

 

If Xander runs the other way then he must cover a distance 2L/3 in time t₂ and the train must cover a distance 

X + L in the same time, so

 

(3) 2L/3 = V_x · t₂  and

(4) X + L = V_t · t₂

 

If we compare equations (1) and (3) we see that t₂ =  2·t₁

Replace t₂ in equation (4) to get:

 

X + L = 2·V_t ·t₁

 

Now replace X with its value in equation (2)

 

V_t·t₁ + L = 2·V_t·t1  ⇒ V_t · t₁ = L

 

Replacing L in equation (1) gives:

 

(V_t · t₁)/3 = V_x · t ₁ ⇒ V_x = V_t/3 = (45 mi/hr)/3

 

V_x = 15 mi/hr