I'm a pure mathematician by trade, but have been teaching myself classical mechanics. I've got to the chapter on Work, Energy and Power and I've found an example that is causing me some problems.
Question
A van of mass 1250 kg is travelling along a horizontal road. The van's engine is working at 24 kW. The constant force of resistance to motion has magnitude 600 N. Calculate:
- The acceleration of the van when it is travelling at 6 m/s.
- The maximum speed of the van.
Answer
Part 1 is fine. We know that $P = Fv$ and so $24000 = 6F$. It follows that $F = 4000$ N. To find the acceleration, we use $F=ma.$ The total resulting force is the traction minus the resistance, in other words $F = 4000 – 600 = 3400$ N. The mass is 1250 kg and so $3400 = 1250a$. It now follows that $a=2.72$ m/s/s.
Part 2 is the part that causes me problems. The maths isn't the problem, it's the way it's used. At maximum speed the acceleration is zero and so the resultant horizontal force will be zero. The book says that $T' = 600$. There are two problems here: I'm used to a prime denoting differentiation. I guess it just means the new tractive force. But the resistance force is working against the direction of motion, so shouldn't we have $T' = -600$?
Running with $T' = 600$, the book then uses $P = Fv$ to get $24000=600v$ and so $v=40$ m/s. I can see that using $T'=-600$ would give a negative velocity, which is clearly untrue.
In short
I don't see why the fact that at maximum speed there will be no acceleration, so the resultant horizontal force will be zero leads to us using $T'=600$ when $T > 0$ was the forwards tractive force and $-600 < 0$ was the resistance force.
Best Answer
I don't think you initially defined $T$, so I'm not entirely sure what it originally used for. The overall idea, however, is that because the newt force is zero (and the resistive force stays the same, i.e. $-600$, then the new forward force has to be $T' = +600$.
Physics is often (e.g. besides high-energy) much more pragmatic than mathematics per se -- one expression of this is the mixed use of things like primes, or superscripts, etc. Similarly, negative signs often aren't tracked quite as closely (as they should be). Because the positive or negative are purely conventional, and it is conceptually clear that the resulting force (and velocity) need to be opposite the resistive force -- then you can just give them the appropriate sign. It is, of course, better to treat them accurately and consistently, but note that this is a common location of hand-waving in introductory level textbooks.