What's the error in my derivations? (impulse/momentum)

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    Derivations Error
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The discussion centers on the relationship between force and momentum, specifically how the integral of force over time relates to momentum. It highlights that the change in momentum is represented by the area under the force versus time curve. A key point raised is the oversight of the integration constant in the derivation, which is crucial for accurately expressing momentum. The correct formulation states that the final momentum equals the initial momentum plus the change in momentum. This clarification is essential for resolving the confusion in the derivation of momentum from force.
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(Change in momentum is the area under a force against time curve.)

7e4c6b89a53d0f0618da45b0241d6c4d.png


(Force is the time derivative of momentum.)

Using separation of varibles, you get Fdt=dp. Integrate both sides, you get that the integral of Force with respect to time is equal to p. This seems to imply that p, momentum, is equal to change in momentum?
 
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Kyuutoryuu said:
Using separation of varibles, you get Fdt=dp. Integrate both sides, you get that the integral of Force with respect to time is equal to p. This seems to imply that p, momentum, is equal to change in momentum?
Look at your first equation. It defines the integral of F (in your sense) as a change in momentum.
 
Kyuutoryuu said:
1e3552f0ba72ecafdafc6b9f80b92d84.png


(Change in momentum is the area under a force against time curve.)

7e4c6b89a53d0f0618da45b0241d6c4d.png


(Force is the time derivative of momentum.)

Using separation of varibles, you get Fdt=dp. Integrate both sides, you get that the integral of Force with respect to time is equal to p. This seems to imply that p, momentum, is equal to change in momentum?

You forgot the integration constant. Momentum after = momentum before + change in momentum.
 

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