Bridging connection between Newton's second law and Work

In summary, to bridge the equations F = ma and m/2(dv2/dx), one can multiply both sides of the equation F = ma by dx and integrate both sides to get 0.5mv^2. To formally differentiate d/dx(v^2), the derivative is 2v(dv/dx).
  • #1
negation
818
0

Homework Statement



I'm trying to bridge F =ma to m/2(dv2/dx). It was shown in the course book I have but there's a huge disconnection in the steps.

The Attempt at a Solution

F =ma = m.(dv/dt) = m(dv/dx . dx/dt) = mv(dv/dx). Where do I take it from here?
 
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  • #2
Formally differentiate d/dx(v^2). What do you get?
 
  • #3
negation said:

Homework Statement



I'm trying to bridge F =ma to m/2(dv2/dx). It was shown in the course book I have but there's a huge disconnection in the steps.

The Attempt at a Solution




F =ma = m.(dv/dt) = m(dv/dx . dx/dt) = mv(dv/dx). Where do I take it from here?

Multiply both sides of your equation by dx to get

F dx = mv dv

Integrate both side, what do you get?
 
  • #4
dauto said:
Multiply both sides of your equation by dx to get

F dx = mv dv

Integrate both side, what do you get?

Both sides?
I could arrive at the conclusion if I were to integrate only the left argument.

∫F(x).dx = ∫ma.dx = ∫m(dv/dt).dx = ∫m(dv/dx . dx/dt) .dx = ∫m(dv/dx . v).dx = ∫m*dv.v = ∫mv.dv = m∫dv . ∫v.dv = m (0.5v2) = 0.5mv2 + C
 
  • #5
rude man said:
Formally differentiate d/dx(v^2). What do you get?

derivative of v2 = 2v(dv/dx)
 
Last edited:

Related to Bridging connection between Newton's second law and Work

What is Newton's Second Law?

Newton's Second Law, also known as the Law of Acceleration, states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. This means that the heavier the object, the more force is needed to accelerate it.

What is Work in terms of physics?

In physics, Work is defined as the product of force and displacement, where the force acts in the same direction as the displacement. It is a measure of the energy transferred to or from an object by an external force.

How is Work related to Newton's Second Law?

According to Newton's Second Law, the net force acting on an object is directly proportional to its acceleration. This means that if a force is applied to an object, it will accelerate and therefore, do work. The work done by a force is equal to the force multiplied by the displacement of the object in the direction of the force.

What is the formula for calculating Work?

The formula for calculating Work is W = F * d * cosθ, where W is the work done, F is the force applied, d is the displacement of the object, and θ is the angle between the force and displacement vectors. This formula is derived from the dot product of force and displacement vectors.

How does understanding the connection between Newton's Second Law and Work benefit scientists?

Understanding the connection between these two principles helps scientists accurately predict and analyze the motion of objects. It allows them to calculate the amount of force needed to move an object a certain distance, or the distance an object will travel when a certain force is applied. This knowledge is crucial in fields such as mechanics, engineering, and physics.

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