Conceptual question about the Kinetic Energy Equation

In summary, kinetic energy, denoted as K, is equal to half of the mass (m) multiplied by the square of the speed (v). This speed, however, is not the same as velocity, as it only represents the magnitude of the velocity vector without any direction information. This means that the expression v=\frac{dx}{dt} is not specifying a particular direction, but rather represents the speed at which an object
  • #1
Xyius
508
4
So as you all know, kinetic energy is..
[tex]K=\frac{1}{2}mv^2[/tex]

The question I have is the followng.

The "v" in this formula isn't really velocity, its speed is it not? Which is just the magnitude of the velocity vector with no direction information. Kinetic energy is a scalar quantity so with this formula am I allowed to write..

[tex]v=\frac{dx}{dt}[/tex]

Since this is just speed, which is the magnitude of the velocity vector, how does this make sense? Couldn't it just as easily be dy/dt or dz/dt? When deriving the kinetic energy formula from Newtons Second Law, you end up with the integral of "Fv" on one side and manipulate v to "dx/dt" to get the work integral.

I guess the main point of my question is, how can v=dx/dt when "v" isn't specifying any particular direction?
 
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  • #2
v2 is the dot product of (v)(v) which is a scalar equal to the square of the magnitude of v. So velocity2 = speed2, but velocity in the expression 1/2 m v2 is a vector and does have a direction.
 
  • #3
rcgldr said:
v2 is the dot product of (v)(v) which is a scalar equal to the square of the magnitude of v. So velocity2 = speed2, but velocity in the expression 1/2 m v2 is a vector and does have a direction.

So basically, only the square of v is scalar? But velocity itself is a vector? Makes sense! Thanks :D
 
  • #4
Consider this:

[tex]K = \frac{1}{2}mv^2 = \frac{1}{2}m (v_x^2 + v_y^2 + v_z^2)
= \frac{1}{2}m \left[ {\left( \frac{dx}{dt} \right)}^2 + {\left( \frac{dy}{dt} \right)}^2
+ {\left( \frac{dz}{dt} \right)}^2 \right][/tex]
 
  • #5


Thank you for your question. You are correct in your understanding that the "v" in the kinetic energy equation represents speed, which is the magnitude of the velocity vector. This means that it does not have a specific direction associated with it. In physics, speed is defined as the rate of change of position, which is represented by the derivative of position with respect to time, or dx/dt.

The reason we use "v" in the kinetic energy equation instead of "dx/dt" is because "v" is a more general representation of speed, while "dx/dt" is specific to a particular direction. In other words, "v" encompasses all possible directions of motion, while "dx/dt" only represents motion in one specific direction.

Additionally, when deriving the kinetic energy equation from Newton's Second Law, we use the integral of "Fv" because it represents the work done by a force in a particular direction. This allows us to calculate the total work done, regardless of the direction of motion. So while "v" may not have a specific direction associated with it, it still represents the speed at which work is being done.

In summary, the use of "v" in the kinetic energy equation is a general representation of speed, while "dx/dt" is specific to a particular direction. Both are valid representations of speed, but "v" is more commonly used in physics because it encompasses all possible directions of motion. I hope this clarifies your question.
 

1. What is the Kinetic Energy Equation?

The Kinetic Energy Equation is a mathematical formula that calculates the amount of energy an object possesses due to its motion. It is represented as KE = 1/2mv², where KE is kinetic energy, m is the mass of the object, and v is the velocity of the object.

2. How is the Kinetic Energy Equation derived?

The Kinetic Energy Equation is derived from the work-energy theorem, which states that the work done on an object is equal to the change in its kinetic energy. By integrating the force with respect to distance, the equation can be derived.

3. What are the units for kinetic energy?

The units for kinetic energy are Joules (J) in the SI system. In the English system, the units are foot-pounds (ft-lb).

4. Does the mass or velocity have a greater impact on kinetic energy?

The velocity has a greater impact on kinetic energy because it is squared in the equation. This means that a small increase in velocity can result in a significant increase in kinetic energy, while a large increase in mass would only result in a minor increase in kinetic energy.

5. Can the Kinetic Energy Equation be used for all types of motion?

The Kinetic Energy Equation can be used for any type of motion as long as the velocity is constant. In cases where the velocity is not constant, such as in cases of non-uniform acceleration, the equation becomes more complex and requires the use of calculus to calculate the kinetic energy.

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