How Do Mass and Speed Affect Kinetic Energy in Cars?

[me_and_you]

Kinetic Energy and cars... Please help

Homework Statement

One car has twice the mass of a second car, but only half as much kinetic Energy. When both cars increase their speed by 6.0 m/s, they then ave the same kinetic energy. What were the original speeds of the two cars.

Homework Equations

Im not sure but i think its:

KE=.5mv_f²-.5mv_i²

The Attempt at a Solution

I manipulated the above equation by making KE multiplied by .5 and then removing the .5 from both masses.This however got me no where so could someone please help?

 

[jgens]

Try writing out the intial and final kinetic energy of each car and then go from there.

 

[thomate1]

Hello,

It seems like you are on the right track with using the kinetic energy equation, but let's break it down further to help you solve the problem.

First, let's define some variables:
m1 = mass of the first car
m2 = mass of the second car
v1 = original speed of the first car
v2 = original speed of the second car
vf1 = final speed of the first car
vf2 = final speed of the second car

Now, let's write out the given information in terms of these variables:
m1 = 2m2
KE1 = 0.5m1v1^2 = 0.5(2m2)v1^2 = m2v1^2
KE2 = 0.5m2v2^2

Next, we can use the given information that both cars increase their speed by 6.0 m/s to write the following equations:
vf1 = v1 + 6.0
vf2 = v2 + 6.0

Now, we can set the final kinetic energies of both cars equal to each other, since they have the same kinetic energy after the speed increase:
KE1 = KE2
m2vf1^2 = m2vf2^2
m2(v1+6.0)^2 = m2(v2+6.0)^2

Finally, we can substitute in the values we know for m1 and m2 to solve for the original speeds of both cars:
m1 = 2m2
2m2(v1+6.0)^2 = m2(v2+6.0)^2
2(v1+6.0)^2 = (v2+6.0)^2
2v1^2 + 24v1 + 72 = v2^2 + 12v2 + 36
2v1^2 - v2^2 + 24v1 - 12v2 + 36 = 0

This is a quadratic equation that can be solved using the quadratic formula. Once you have solved for v1 and v2, you can then use these values to calculate the original speeds of the cars (v1 and v2) before they increased their speed by 6.0 m/s.

I hope this helps! Let me know if you have any further questions.