A climbing rope exerts a force given by F=−kx−cx2. Find an expression for c such that when the rope is stretched a distance d its potential energy is twice what it would be if the rope were an ideal spring with F=−kx.

Explanation:

For the climbing rope, the force is given by,

Now, we know that for any conservative force,

(one dimensional case)

where, V = potential energy

Therefore, the potential energy of the rope is given by,

a is a constant, which can be taken to be zero depending on our choice of zero potential. Hence,

Similarly, we can find the potential energy of an ideal spring,

Now, it is given that at x = d,

or,

or,

The two equations are:

y = 5*x + 3

y = 5.5*x + 2

Step-by-step explanation:

By definition:

speed = distance/time

it can be rewritten as:

time = distance/speed

During the first half of the race Ben ran at a rate of 5 miles per hour for x hours and walked 3 miles. Let's call y, the number of miles in half the race; then he ran y-3 miles. The time he spend was:

x = (y-3)/5

Analogously, for the second half:

x = (y-2)/5.5

These two equations can be rewritten as

y = 5*x + 3

y = 5.5*x + 2

Which means that:

5*x + 3 = 5.5*x + 2

3-2 = 5.5*x -5*x

1 = 0.5*x

x = 2 hours

When you are solving for the variables in a situation represented by two equations you can isolate the same variable in both equations and equate the two terms, getting a new equation with only one variable. Another option is to isolate one variable in one equation and replace it in the other one.