Question:

A car drives straight down toward the bottom of a valley and up the other side on a road whose bottom has a radius of curvature of 115 m. At the very bottom, the normal force on the driver is twice his weight. At what speed was the car traveling?

Source: Giancoli, Douglas C., Physics: Principles with Applications, 7th Edition, 2014.

Quick Answer:

$33.6 \textrm{ m/s}$

### Transcript for this Giancoli solution

This is Giancoli Answers with Mr. Dychko. Here's a picture of what's happening with this car traveling down through this valley. At the bottom of the curve, it's going through a circle of radius 115 meters and has some tangental velocity, or some speed, this way. The free-body diagram looks like this, with the normal force upwards, with an arrow that's twice as long as the gravity downwards because we are told the normal force is 2 times gravity. And so we can draw, write Newton's second law, with these forces; we have the up forces minus the down forces equals mass times acceleration. And so, normal force minus gravity equals*ma*. And then make substitutions for each of these things; we can say normal force is 2 times gravity coz we are told that by the question; gravity is mass times gravitational field strength,

*g*, and then acceleration since it's accelerating in a circle going at constant speed, the acceleration is centripetal acceleration so we can write

*v squared*over

*r*instead of

*a*. So we now make substitutions into Newton's second law from here. So we have normal force then is 2 times force of gravity so that's 2 times

*mg*, and force of gravity is

*mg*. And then we have mass copied and then times acceleration,

*v squared*over

*r*. The

*m*'s cancel everywhere; and divide both sides by

*m*and 2

*g*minus

*g*makes

*g*and we have

*g*is

*v squared*over r. And then we solve for

*v*by multiplying both sides by

*r*and take the square root of both sides and we have the speed must be the square root of the radius of curvature times

*g*. So square root of 115 meters times 9.8 meters per second squared which is 33.6 meters per second.