Calculus Kinetic Energy Question?

Question

The kinetic energy, E, of a moving object is directly proportional to the square of its velocity, v. The velocity of a certain object is 3 meters per second and its kinetic energy is 15 joules. What velocity will be necessary in order for this object's kinetic energy to equal 60 joules?

Answer 1

You can do this the brute force way, by finding k:

 

If E is proportional to v² then you can say E = kv², with k being some constant.

 

So based on the first fact, we know that 15 = k * 3² = 9k.  That allows us to solve for k:

 

15 = 9k

5/3 = k

 

Now use this to solve the problem:

 

E = 5/3 v²

 

60 = 5/3 v²

60 / (5/3) = v²

36 = v²

6 = v

 

 

Or...  You can notice that the kinetic energy at the second point, 60, is 4 times the kinetic energy at the first point, 15.  Since the kinetic energy is proportional to the SQUARE of the velocity, that means that the velocity must have increased by a factor of 2 times the original velocity (since 4 is the square of 2).  So the velocity would have to be 2 times 3 to give a kinetic energy of 4 times 15, and the answer is 2 * 3 = 6.