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Question Video: Identifying the Equation for Newton’s Second Law with a Variable Mass Mathematics

Complete the following equation: 𝐹 = d/d𝑡 (mass × _).

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Video Transcript

Complete the following equation: 𝐹 equals d by d𝑡 of mass times what.

When we think about equations that describe forces acting on a body, we tend to think about Newton’s second law of motion for constant mass. The magnitude of the acceleration of a body depends on the magnitude of a force and the mass of the body. And the formula we use is 𝐹 equals 𝑚𝑎. However, we can also think about Newton’s second law of motion in terms of the rate of change of the momentum of the body. That is, force is equal to the derivative of momentum with respect to time.

But of course, the momentum of a body is given by 𝑚𝑣, where 𝑚 is its mass and 𝑣 is its velocity. So we can, alternatively, calculate the force by finding the derivative of mass times velocity with respect to time. And with that in mind, we can now complete the equation in our question 𝐹 is d by d𝑡 of mass times what. Well, we know it’s the derivative with respect to time of 𝑚𝑣, of mass times velocity. And so the missing word is velocity.

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