Given a body's initial and final velocity, find its maximum acceleration for a given power and mass

Question

Stan is driving his $35 (metric) tonne$ truck on a horizontal road. Stand accelerates from $50 km h^-1$ to $65 km h^-1$, which is his maximum speed at $500kW$ power output. Find the maximum acceleration of the truck, assuming the total resistance is constant.

Suppose a 35000kg truck is driving along a horizontal road.

The truck accelerates from $13.889ms^{-1}$ to $18.0556ms^{-1}$.

$18.0556ms^{-1}$ is the truck's maximum speed at 500,000W power output.

Assuming the total resistance is constant, find the maximum acceleration of the truck.

The answer is $0.24ms^{-2}$ but I don't seem to be able to get to it.

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OK, here are my thoughts, imagine the two cases at the start and end.

At the end the truck is at its maximum velocity so its velocity is constant. Therefore friction = driving force.

The driving force can be calculated by

$P = Fv$

As friction is constant, we therefore know the friction at the start.

Mechanical energy is conserved

so $work done = final KE + lost energy - initial energy$

Calculate the resultant using

$F_{Resultant}= F_{driving}-F_{friction}$

Finally I imagine Newton's second law is need

$F=ma$

to find acceleration assuming the acceleration is greatest initially

I expect the solution will involve the conservation of energy as well. I have given the exact wording of the question in addition to my interpretation of the wording.

Answer 1

It's a long time since I've done any physics, so I hope the community will correct me if something is wrong here.

You're right that at the end of the acceleration we have: $$D=R$$ where $D=\frac{P}{v_f}$ is the force of the motor, and $R$ is friction. We find that $R=\frac{500'000}{18.0556}\approx 27'692$ N. At the start of the acceleration the driving force is maximum when $D=\frac{500'000}{13.889}\approx 35'999$ N.

Thus by Newton's 2. law, the greatest acceleration of the truck is $$a=\frac{D-R}{m}=\frac{35'999-27'692}{35'000}\approx 0.237\frac{m}{s^2}$$

Answer 2

Asking the question seemed to clear my own thoughts and has enabled me to answer it.

I have inferred from the question that the power output is constant at $500kW$.

I have already down the conversions to SI units for the mass and velocities.

As the body is decelerating, its greatest acceleration is at the start. Also, I am assuming the power output of the engine is constant.