The Problem:
A freight train 1/4 km long takes 20 seconds to pass a signal. The train is decelerating at a constant rate, and by the time the rear truck has passed the signal it is moving 10 kilometres per hour slower than it was when the front of the train passed the signal. Find the deceleration in kilometre-hour units, and the speed at which the train is moving when the rear truck has just passed the signal.
A freight train 1/4 km long takes 20 seconds to pass a signal. The train is decelerating at a constant rate, and by the time the rear truck has passed the signal, it is moving 10 km/h slower than it was when the front of the train passed the signal. Find the deceleration in km/h² and the speed at which the train is moving when the rear truck has just passed the signal.
Solution:
Step 1: Define Variables
- Let u = initial speed of the train (in km/h).
- Let v = final speed of the train (in km/h).
- Let a = deceleration (in km/h²).
From the problem:
v = u - 10
Step 2: Relate Distance, Speed, and Time
Using the equation of motion:
L = ut + (1/2) a t²
Substitute the known values:
0.25 = u (1/180) + (1/2) a (1/180)²
Simplify:
0.25 = u/180 + a/64800
Step 3: Relate Final Speed to Deceleration
Using the equation of motion:
v = u + a t
Substitute v = u - 10 and t = 1/180:
u - 10 = u + a (1/180)
Simplify:
-10 = a/180
Solve for a:
a = -1800 km/h²
Step 4: Solve for Initial Speed (u)
Substitute a = -1800 into the distance equation:
0.25 = u/180 - 1800/64800
Simplify:
0.25 = u/180 - 1/36
Add 1/36 to both sides:
0.25 + 1/36 = u/180
Convert 0.25 to a fraction:
5/18 = u/180
Solve for u:
u = 50 km/h
Step 5: Solve for Final Speed (v)
Using v = u - 10:
v = 50 - 10
v = 40 km/h
Final Answer:
- Deceleration (a): 1800 km/h²
- Final speed (v): 40 km/h
