PROBLEM:
A subway train travels 400ft between two stations. It starts from rest and accelerates at the rate of 8ft/sec^2 until it's velocity reaches 20ft/sec. It then moves at this constant velocity for awhile and then decelerates to rest at the rate of 12ft/sec^2. Find the total time between the stations.
QUESTIONS:
I'm lost on how to approach this problem. I'm not for sure if I should set two acceleration expressions up and use integration to work backwards. Then somehow utilize both to arrive at (t).
I started with 8 as acceleration and used integration to get v as shown below.
(a) = 8
(v) = 8t + 20 NOTE: This seems wrong because initial velocity is technically 0ft/sec
400 = ?
Again, this can't be so and I'm stuck on how and where I should start implementing 20ft/sec and the other deceleration rate which is 12ft/sec^2. Can someone get me pointed in the right direction without giving me the complete answer.
Best Answer
I see someone has already answered this.
I tend to tell students to break up the problems according to differing acceleration/velocities and build an expression for the total time in terms of the known variables. Also if you use integration the correct expression for v would be $$ v(t) - v(0) = \int_{0}^t a(t) dt $$ Where you would use the fact that your initial condition is $v(0) = 0 $ to arrive at $$ v(t) = 8t + v(0) = 8t $$ Then you might further integrate using the initial condition $x(0) = 0$ to obtain in a similar fashion $$ x(t) - x(0) = \int_0^t v(t) = \frac{1}{2}at^2$$ $$ x(t) = \frac{1}{2}at^2 + x(0) = \frac{1}{2}at^2 $$ Which is the usual distance equation for a zero initial velocity/position.
However this is over complicating the problem because we already would have at our disposal the equations for constant acceleration: \begin{eqnarray*} x(t) &=& x_0 + v_0 t + \frac{1}{2}at^2 \\ v(t) &=& v_0 + at \\ v^2(t) &=& v^2(t_0) + 2 a \Delta x \\ \end{eqnarray*} But yes the previous post is correct, simply separate the trip into three sections with different accelerations and solve for the individual times