Problem 2/101 Page 59, Engineering Mechanics - Dynamics, Meriam and Kraige, 4th Edition
Given: A train enters a curved track with a initial speed of v0. The train slows down at a constant rate such that the speed decreases by to v1 at time t1. At time tc the magnitude of the acceleration of the train is ac.
Find: The radius of curvature of the track at the position occupied by the train at time tc.
Parameter values:
v0 = 100 km/hr
t1 = 12 s
v1 = 50 km/hr
tc = 6 s
ac = 2 m/s2
a = at t + an n
at = dv/dt = (v1-v0)/t1 = -1.157 m/s2
an = v2/r
Separating variables
dv = (v1-v0)/t1 dt
Integrating both sides:
v = (v1-v0) t/t1 + C
Noting that at t=0, v=v0, permits the evaluation of C.
v = (v1-v0) t/t1 + v0
v(tc) = (v1-v0) tc/t1 + v0 = 75 km/hr = 20.8 m/s
Noting that the magnitude of the acceleration is the square root of the sum of the squares of the components:
| a | = { at2 + an2 }1/2
Evaluating this at t=tc and squaring both sides:
ac2 = at2 + { v(tc)2/r }2
Solving for 1/r:
1/r = { ac2 - at2 }1/2 / v(tc)2
Using the numerical values obtained above:
r = 266 m