How Is Kinetic Energy Calculated from Force and Displacement?

[cereal turtle]

Homework Statement

If the resultant force acting on a 2.0kg object is equal to (3i + 4j) N, what is the change in kinetic energy as the object moves from (7i -8j)m to (11i -5j)m

Homework Equations

w = Fd
Ek = 1/2mv²

The Attempt at a Solution

I assumed that Ek = w so I tried to solve it by getting the change in displacement and then getting the dot product using the given force (did not use the mass at all)

d= (11i -5j) - (7i -8j)
= (4i +3j)

w= (3i +4j)(4i + 3j)
= (12i + 12j)

|w| = 17 J

... this answer was totally wrong, as the answer should be 24 J.

I can't really grasp the relationship with force and kinetic energy.. would appreciate some help. thank you.

 

[BeBattey]

Your dot product is incorrect. Remember it's also called the scalar product, because it produces a scalar, not a vector.

The dot product of two vectors, [a,b] and [c,d], is ac+bd, a simple number. In your case you have everything else correct.

Gotta kick that habit of tacking i and j back on there after evaluating the dot product!

You can also see how this is wonky because you're saying w, work, is a vector quantity, which energy is not!

 

[cereal turtle]

Ohhh! thanks so much! so i really didn't need the 2.0kg?

 

[BeBattey]

No, you really don't. The force over the distance is the work done, or change in kinetic energy. If you have a very massive object, the change in kinetic energy would be the SAME, but the resulting velocity would be lower!

 

[HalfLostAndSearching]

Hello,

It seems like you are on the right track with your approach. The equation Ek = 1/2mv^2 is used to calculate the kinetic energy of an object in motion, while w = Fd is used to calculate the work done on an object by a force. In order to find the change in kinetic energy, we need to use the work-energy theorem, which states that the net work done on an object is equal to the change in its kinetic energy.

So in this case, we can use the given force and displacement to calculate the work done on the object, and then use that value to find the change in kinetic energy.

First, let's find the displacement vector:

d = (11i - 5j) - (7i - 8j)
= (4i + 3j)

Next, we can calculate the work done using the dot product of the force and displacement vectors:

w = F * d
= (3i + 4j) * (4i + 3j)
= 12i + 9j

Now, we can use the work-energy theorem to find the change in kinetic energy:

Ek = w
= (12i + 9j)
= 15 J

Therefore, the change in kinetic energy is 15 J. It's important to note that kinetic energy is a scalar quantity, so we don't need to worry about the direction of the vectors in this case.

I hope this helps clarify the relationship between force and kinetic energy. Let me know if you have any further questions.

Best,

Scientist