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MHD93
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Why is dp/dt = F even if the mass is changing, can it be derived from Newton's second law?
Derivative of Momentum: Why is dp/dt = F?
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In summary: However, if you are asking whether F = ma is a special case of the 2nd law, then the answer is yes. In summary, dp/dt = F is the mathematical formulation of Newton's second law, which states that force is proportional to acceleration. This equation can be derived from the original formulation of F = ma, but only when the mass is constant, as mass is the proportionality constant in this case. Therefore, F = ma is a special case of the second law, where the rate of change of mass is equal to 0.
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tiny-tim
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Hi Mohammad! 
That is Newton's second law …
We can only use "F = ma" when the mass is constant.
(constant mass is the exception)
Mohammad_93 said:
That is Newton's second law …
force = rate of change of momentum.
We can only use "F = ma" when the mass is constant.
(constant mass is the exception
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MHD93
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Newton's second law says that force is proportional to acceleration, meaning that mass is the proportionality constant
Isn't that true?
Isn't that true?
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olivermsun
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It doesn't really matter how you write it. The original formulation is F = dp/dt, but if mass is constant you can take it outside the derivative. If not, then you will need to account for the momentum carried into or out of the system by the mass (change).
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DrBloke
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Another may of looking at it, but reiterating what the others have said, is as follows:
p=mv
F=dp/dt (Newton's second law states force equals rate of change of momentum)
F=d(mv)/dt
=mdv/dt +vdm/dt (equation 1)
if the mass is constant then the rate of change of mass, dm/dt=0 so this becomes
F=mdv/dt
=ma (equation 2)
p=mv
F=dp/dt (Newton's second law states force equals rate of change of momentum)
F=d(mv)/dt
=mdv/dt +vdm/dt (equation 1)
if the mass is constant then the rate of change of mass, dm/dt=0 so this becomes
F=mdv/dt
=ma (equation 2)
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Mohammad_93 said:
As others may have stated, dp/dt = F_net IS the mathematical form of the 2nd law, thus it can't be derived from itself.
FAQ: Derivative of Momentum: Why is dp/dt = F?
What is the definition of momentum?
Momentum is a measure of an object's motion, calculated as the product of its mass and velocity. It is a vector quantity, meaning it has both magnitude and direction.
What is the formula for momentum?
The formula for momentum is p = mv, where p is momentum, m is mass, and v is velocity. It can also be written as p = mΔv, where Δv is the change in velocity over time.
What is the relationship between force and momentum?
Force is directly proportional to the rate of change of momentum. This means that if force is applied to an object, its momentum will change at a constant rate. Mathematically, this can be represented as F = dp/dt, where F is force, p is momentum, and t is time.
How is the derivative of momentum calculated?
The derivative of momentum is calculated using the chain rule of calculus. This involves taking the derivative of the product of an object's mass and velocity, which results in the derivative of momentum being equal to the mass multiplied by the derivative of velocity with respect to time.
Why is dp/dt equal to F?
This is because, as mentioned earlier, force is directly proportional to the rate of change of momentum. In other words, the derivative of momentum is equal to the applied force. This relationship is fundamental in understanding the dynamics of objects in motion and is a key concept in physics.