More A.P. French Mechanics

A. P. French has a knack for creating simple problems I can't solve. I would appreciate some suggestions. (This is 6-8, for those who have his book on mechanics.)

In part 1, a car, m=10^3 kg, is traveling at 28 m/s on a horizontal straight road. The driver sees a tree 100 m away, and with a reaction time of .75 seconds, applies the brakes and stops 9 m short of the tree. What is the decelerating force?

This was not hard, I got -5600 N for the force. F/mg = 4/7, which is important later.

Part b is the part I can't seem to solve. I think I have it right, but it doesn't agree with the answer in the book.

B) Same situation as before, same decelerating force as before, but this time it is on a grade of arcsin (1/10). What speed does the car hit the tree?

For the sum of the X forces, I get f_grav - f_fric = f_car, or since F_fric/mg = 4/7,

mg sin A - (4/7)mg cos A = ma

980 - 5600 cos A = 10^3 a, or a = 4.59

I use v^2 = 28^2 - 2(4.59))(79), and get v^2 = 58.78.

(79 m is the distance to the tree, after subtracting the reaction time distance of D = 28 m/s (.75) = 21 from the total distance of 100 m.)

This all seems right to me, but Mr. French gives the value of v^2 as 42.

Could someone point out where I have gone wrong? (This is not homework, I am using the French text for extra reading and problems.)

Thank you,
Dot

 

Thanks. I don't understand this though. Wouldn't the normal force be less on the incline? This is the same value for the frictional force as one would have on a perfectly horizontal surface. And it also doesn't deliver the answer in the book, which I am beginning to think might be wrong.

Dot

 

Yes, this makes sense, and is how I started out, but when I didn't get the correct answer, I tried other ways, this seemed the most sensible. However, nothing I have done has worked out the way French thinks it should.

It's frustrating to feel that I am missing some crucial insight in what really seems like a trivial problem.

Dot

 

Sure. Thanks for your time and help with this.

A car of mass 10^3 kg is traveling at 28 m/sec (a little over 60 mph) along a horizontal straight road when the driver suddenly sees a fallen tree blocking the road 100 m ahead. The driver applies the brakes as soon as his reaction time (0.75 sec) allows and comes to rest 9 m short of the tree.

(a) Assuming constant deceleration caused by the brakes, what is the decelerating force? What fract is it of the weight of the car (take g = 9.8 m/sec^2)?

(b) If the car had been on a downward grade of sin^{-1} (1/10) with the brakes supplying the same decelerating force as before, with what speed would the car have hit the tree.
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The answer to (b), which is what is causing me so much grief, is given in the book as the square root of 42, approximately 15 mph.

Thanks again,
Dot