Question

A locomotive with a (constant) power capability of 1.8 MW can accelerate a train from a speed of 13 m/s to 30 m/s in 8.8 min. (a) Calculate the mass of the train. Find (b) the speed of the train and (c) the force accelerating the train at 3.1 min after it started accelerating from the speed of 13 m/s. (d) Find the distance moved by the train during the 8.8 min interval that it took to reach the speed of 30 m/s. Note: This Problem involves constant power, not constant force.

Answer 1

Since the question explicitly says at the end that we aren't dealing with constant force, it's important to realize that we can't assume constant acceleration, meaning all our equations of linear motion go out the window.

For part a), by conservation of energy, any work put into accelerating the train is converted into kinetic energy (assuming there's no friction). In this case, the work corresponds to the CHANGE in kinetic energy - the amount of energy it takes to get from 13 m/s to 30 m/s. Once you realize this, it's easy to come up with an equation - so here's what we have:

\[P=\frac{W}{t}\]\[P=\frac{KE_2-KE_1}{t}\]\[P= \frac{\frac{1}{2} mv_f^2-\frac{1}{2}mv_2^2}{t}\]\[m=\frac{2tP}{v_f^2-v_i^2}\]\[m=\frac{2(8.8 \times 60)(1.8 \times 10^6)}{30^2-13^2}\]\[m=2.6 \times 10^6\]
The mass of the train is 2.6 x 10^6 kg.

For part b), you're actually going to be using the exact same equation, but solving for vf rather than m (the fact that you're told at the end that the question is about constant power, not force, should be a hint to that). Try it yourself and see what you get!

For part c), you're again using the power equation, but in a slightly different form. Look at this:

\[P=\frac{W}{t}\]\[P=\frac{F \Delta d}{t}\]\[P=Fv_f\]
You've just found vf in part b), so now you can use that to solve for the force being applied! Incidentally, this is why we can't assume constant force: if you look at the second line of the equation above, you can have many combinations of force and distance to produce the same power in a given period of time.

Finally, for part d), you can use this same equation again - just take it one step further:
\[P=\frac{W}{t}\]\[P=\frac{F \Delta d}{t}\]\[P=\frac{ma \Delta d}{t}\]\[P=\frac{m \Delta v \Delta d}{t^2}\]
Note that this is using average force, and therefore average acceleration - but that's totally ok in this situation because we're calculating it from given velocities and time. Use this last equation to solve for Δd, because you know the values of every other variable!

Answer 2

Thank you :)