# Question Video: Identifying the Equation for Newtonโs Second Law with a Variable Mass Mathematics

Complete the following equation: ๐ = d/d๐ก (mass ร ๏ผฟ).

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### Video Transcript

Complete the following equation: ๐ equals d by d๐ก of mass times what.

When we think about equations that describe forces acting on a body, we tend to think about Newtonโ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 ๐ equals ๐๐. However, we can also think about Newtonโ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 ๐๐ฃ, where ๐ is its mass and ๐ฃ 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 ๐ is d by d๐ก of mass times what. Well, we know itโs the derivative with respect to time of ๐๐ฃ, of mass times velocity. And so the missing word is velocity.