Tough physics question please help for best answer points!?

Question

A high speed train traveling at 161 km/hr rounds a bend and is shocked to see that a locomotive has entered the tracks going the same direction .676 km ahead of it. The locomotive is traveling at 29.0 km/hr. The engineer of the high speed train applies the brakes. What must be the magnitude of the constant deceleration of the train so that a collision is just avoided?

Some equations that may help:

Final Velocity = Original Velocity + acceleration(time)

position = (1/2)acceleration(time squared) + original velocity(time)

Final velocity squared = original velocity squared + 2(acceleration)(position)

or

Vf = Vo + at

x = (1/2)at^2 + Vot

Vf^2 = Vo^2 + 2ax

Answer 1

Relative to the train in front, we are going at (161 - 29)km/h

Try plugging in the numbers for the 3rd equation in meters and seconds:

0 = ((161 -29)*1000 / 3600)^2 - 2 deceleration * 676

Answer 2

its gotta be a really really high magnitude or else they gon crash

Answer 3

write the equations of motion for the two trains in the form:

x(t)=x0+v0t+1/2at^2

x(t)=position at any time t, x0=initial position, v0=initial speed, a is accel and t is time

for the locomotive, x0=0.676 km =676m, v0=29km/hr = 8.06 m/s, a=0

for the high speed train, x0=0, v0=161km/hr=44.7 m/s and a is to be determined

we know the distance between the trains will be x(loco)-x(high) and this is:

x(loco)-x(high)=676+8.06t-(44.7t+1/2 at^2)

=676-36.7t-1/2 at^2

but we know that the accel of the high speed train is given by:

(vf-v0)/t=a; since vf for the high speed train is zero, we know that a is -v0/t=-44.7/t

we substitute this into the above to get:

x(loco)-x(high)=

676-36.7t-1/2(-44.7/t)t^2 =

676-36.7t+22.3t

since a collision is averted if x(loco)-x(high) is just zero, we set this to zer to find that the high speed train must slow down in a time:

676-36.7t+22.3t=0 to t=47s

if the high speed train comes to rest in 47 secs, its accel was 44.7/47=0.96 m/s/s

note added in editing:  one answer above will yield essentially the same answer but does this problem in a very elegant and compact way...well done!