A train traveling at v0 = 30.0 m/s begins to brake by applying a velocity-dependent instantaneous acceleration a(v) = α /(v u) m/s2, where α = – 20.0 m2/s3, v is the instantaneous velocity of the train, and u = 0.2 m/s. Determine the distance traveled by the train until it comes to a complete stop.
To determine the distance traveled by the train until it comes to a complete stop, we need to integrate the velocity function over the time it takes for the train to stop.
First, let's find the time it takes for the train to stop. We know that the acceleration is velocity-dependent, given by a(v) = α /(v u). In this case, α = -20.0 m^2/s^3 and u = 0.2 m/s.
When the train comes to a complete stop, the velocity will be zero. So, we can set v = 0 in the acceleration equation and solve for v.
a(v) = α /(v u)
0 = α /(v_stop u)
0 = -20.0 / (v_stop * 0.2)
0 = -100 / v_stop
Solving for v_stop, we find:
v_stop = 0 m/s
Therefore, the time it takes for the train to come to a complete stop is zero.
Now, let's find the distance traveled by the train. To do this, we need to integrate the velocity function over the time interval from t = 0 to t = t_stop.
Since the time it takes for the train to stop is zero, the distance traveled by the train until it comes to a complete stop is also zero.
In summary, the distance traveled by the train until it comes to a complete stop is zero.