### Video Transcript

A body with variable mass moves in a straight line. At time ๐ก seconds, its mass is given by ๐ equals seven times the quantity ๐ก plus four kilograms. And its displacement from a fixed point on the line is given by ๐ equals one-quarter times the quantity ๐ก squared minus two ๐ก plus 10 meters. Calculate the change in the bodyโs momentum between ๐ก equals 10 seconds and ๐ก equals 13 seconds.

Weโll call the time values at either end of our interval of interest, ๐ก equals 10 seconds and ๐ก equals 13 seconds, ๐ก sub ๐ and ๐ก sub ๐, respectively. We want to solve for the change in the bodyโs momentum over this time interval. Weโll call that change ฮ๐ป. To help us solve it, weโre given the mass of the object as a function of time as well as its displacement, ๐ , also as a function of time.

Letโs start our solution by recalling the mathematical equation for momentum. The momentum of an object with mass is equal to the objectโs mass times its velocity ๐ฃ. In our case, we want to solve for a change in momentum, ฮ๐ป. Based on our expression for momentum, we can write this as the magnitude of the difference between the final mass times the final velocity and the initial mass times the initial velocity. Weโll need velocities in order to calculate ฮ๐ป. But weโre given displacement ๐ in our problem statement. The two quantities are related through the expression: ๐ฃ is equal to the time derivative of displacement ๐ .

In order to eventually solve for ๐ฃ sub ๐ and ๐ฃ sub ๐, letโs first find out what ๐ฃ as a function of ๐ก is. Weโll do this by taking the time derivative of ๐ . Plugging in for our displacement ๐ , we calculate the time derivative to be one-quarter times the quantity two ๐ก minus two meters per second. Factoring out a two from inside the parentheses, we now have our velocity as a function of time. We recall that weโre specifically interested in our particleโs velocity at ๐ก equals ๐ก sub ๐ and ๐ก equals ๐ก sub ๐. And likewise, weโre interested in our particleโs mass at those two times.

We can write ๐ sub ๐ times ๐ฃ sub ๐ as seven times 10 plus four kilograms multiplied by one-half times 10 minus one meters per second. This is equal to 49 times 9 kilograms meters per second. So thatโs ๐ sub ๐ times ๐ฃ sub ๐.

Moving on to ๐ sub ๐ ๐ฃ sub ๐, thatโs equal to seven times 13 plus four kilograms multiplied by one-half times 13 minus one meters per second. This is equal to 42 times 17 kilograms meters per second.

With this plugged in to our expression, weโre now ready to calculate ฮ๐ป, the change in momentum. Entering these values on our calculator, we find itโs 273 kilograms meters per second. Thatโs the change in momentum of this object over the given time interval.