# Thread: Mechanics problem: speed of car up a slope

1. ## Mechanics problem: speed of car up a slope

The following is a question from an old UK A-Level Applied Mathematics textbook. I have tried to solve this and agree with the book answer for the first part of the question, but for the second part of the question I get a different answer to the book. Could someone please help me understand what I have done wrong, or whether the book answer is incorrect.

A car of mass 1500 kg has a maximum speed of 150 km h-1 on a level road when the engine is exerting its maximum power of 200 kW. Find the resistance to motion at this speed. If this resistance is proportional to the speed of the car, find the maximum speed of the car up a road inclined at arcsin(0.1) to the horizontal.

This is my answer to the question:

Let F be the force acting on the car and R be the resistance to motion.

At the maximum speed there is no acceleration; therefore, the forces acting on the car are in equilibrium. Therefore F = R. We are given

P = 200 x 103 W, m = 1500 kg, v = 150 km h-1 = 125/3 m s-1

P = Fv, therefore R = 4 800 N

Now for the second part:

When travelling at any speed v, R=kv, so k = R/v = 4800 / (125/3) = 576/5

When travelling up the hill at maximum speed V

F = P/V = 200 x 103/V and R = 576V/5

There is no acceleration, so if F is the force acting on the car

F - 1500g (0.1) = R
200 x 103/V - 1470 = 576V/5
576V2/5 + 1470V - 200 000 = 0

so

The book gives the answer as 13.3 m s-1. What am I doing wrong?

Many thanks

2. ## Re: Mechanics problem: speed of car up a slope

You subtracted the vertical component of gravity. You need the horizontal component. That's cos (arcsin (0.1))=sqrt (99)/10

3. ## Re: Mechanics problem: speed of car up a slope

Nm, I think the book is just wrong and your answer is correct. If you use the orthogonal component of gravity that I suggested in my post above, you wind up with 12.45 m/s, which also is not what the book has. So, I think the book is just wrong.

4. ## Re: Mechanics problem: speed of car up a slope

Thank you. I resolved forces parallel to the incline, so it is the vertical component of gravity that I needed. Glad you think my answer is correct!