How to find the maximum velocity of two objects that travel from a point #A# to a stopping point #B# in the same amount of time?
A train passes through a station #A# and travels for #10\ km# at a constant speed of #60\ \frac{km}{h}# , then it is uniformly decelerated for #3\ km# until stopping in station #B# . A second train leaves from the same station #A# simultaneously with the passage of the first and travels before uniformly accelerated motion and then of uniformly decelerated motion until stopping at station #B# at the same instant of the first train. What is the maximum speed reached by the two trains?
A train passes through a station
1 Answer
Train 1's maximum velocity =
Train 2's maximum velocity =
Explanation:
Train 1's maximum velocity is obviously
Train 2 will not be as easy. In fact we need to study train 1's data to get useful information to find train 2's maximum velocity. The time is the same for both, so getting train 1's time will help.
Train 1 went 10 km at
Then train 1 put on the brakes, decelerating uniformly to a stop in 3 km. What was the acceleration,
It is unusual to see an acceleration with
For the time to come to a stop, use the kinematic formula
OK, total time for either train is
If train 2 is to arrive at the same time as train 1, it must make the trip with an average speed of
Let
Using the kinematic formula
During the accelerating and decelerating phases, we have these 2 formulas
Solving both for
Let's get both accelerations on the left side and both times on the right.
So we have
It looks like if we double
I now see a path to the answer. Much of the work above was necessary, and perhaps interesting, so I will leave it there. If the accelerating part of the trip has an average speed
then since the acceleration is uniform,
I multiplied
Picture a velocity time chart for the acceleration phase (or the deceleration phase). It is a right triangle.
Now consider doing the entire trip at a speed of
So the answer is
I hope this helps,
Steve