Max Steepness Calculation for City Road Design: Physics Problem Solution

[anightlikethis]

1. The problem: A city planner is working on the redesign of a hilly portion of a city. An important consideration is how steep the roads can be so that even low-powered cars can get up the hills without slowing down. It is given that a particular small car, with a mass of 1060 kg, can accelerate on a level road from rest to 20 m/s (72 km/h) in 14.0 s. Using this data, calculate the maximum steepness of a hill.
atement, all variables and given/known data

Homework Equations

The Attempt at a Solution

 

[rmorelan]

I am pretty new to the forum, but I think you actually have to provide some sort of reasonable attempt to the problem before people will normally reply to you.

as a hint here you have enough information to determine the car's acceleration, you know its mass you you can compute the maximum force of the engine. Then you just need to find for the hill the component of the force that would be pulling the car down the hill.

 

[hage567]

Yes, as rmorelan pointed out, you do need to show some work in trying to solve the problem. For starters, have you tried drawing a free body diagram?

 

[anightlikethis]

Yes okay
Well this is all I've got:
I drew a free body diagram and I think that there would be a car accelerating the car up the ramp of 1.43 N(because that is 20/14)*1060kg=1514.21 and gravity pushing it down in a line perpendicular to the ramp at 1060*9.8=10385. I assume I should fsomehow find theta when a=0, but I have no idea how.

 

[rmorelan]

Yes okay
Well this is all I've got:
I drew a free body diagram and I think that there would be a car accelerating the car up the ramp of 1.43 N(because that is 20/14)*1060kg=1514.21 and gravity pushing it down in a line perpendicular to the ramp at 1060*9.8=10385. I assume I should fsomehow find theta when a=0, but I have no idea how.

Looks like you are making progress, but there are a couple of things. by the way this is why they like equations :)

First you must find the acceleration of the car. I am not sure what you are doing for that but the basic equation is

v2 = v1+at

and v1=0 in this case, and you know the car takes 14 seconds to get to 20m/s. So the accelration is 20/14 = 1.43 m/s2

You also know that force equal mass * acceleration (F=ma). You know a now, and you are given mass, so you can find the maximum force this engine can output. This is important, as if on the hill the component of the gravitational force that pulls the car down the hill is greater than the maximum force the engine can produce the car will slow down as it goes up the hill.

Notice the question specifically tells you the car must be able to get up the hill WITHOUT slowing down. Now you have a component problem that is not too hard. You are not setting a=0 anywhere.

that should get you started :)

 

[mybsaccownt]

I assume I should fsomehow find theta when a=0, but I have no idea how.

yes,

but...the total acceleration

in other words, when the sum of the acceleration provided by the car and the acceleration due to gravity (in the direction you're interested in) equal 0

 

[anightlikethis]

Gosh I am sorry. I am really bad at this.
I have the force that is due to gravity and against the car, but how can I get the steepness from that?

 

[hage567]

The force of gravity down the hill should have a $$\sin\theta$$ in it, where $$\theta$$ is the steepness of the hill. So you can then isolate and solve for $$\theta$$.