### Video Transcript

A body of mass seven kilograms is moving in a straight line. Its position vector at a time π‘ is given by the relation π as a function of π‘ equals π‘ squared plus five quantity in the π direction plus π‘ cubed plus π‘ quantity in the π direction, where the magnitude of π is measured in metres and π‘ in seconds. Determine the bodyβs momentum after two seconds.

We can record the bodyβs mass of seven kilograms as π. The time value weβre interested in β two seconds β weβll call π‘ sub two. Weβre given the position of our object as a function of time β π as a function of π‘ β and based on that want to solve for its momentum, which weβll represent by the letter capital π».

To begin on our solution, letβs recall the mathematical relationship describing momentum. The momentum of an object with mass is equal to that objectβs mass times its velocity. We know that an objectβs velocity is defined as its change in position over time β ππ ππ‘.

Combining these relationships, we can write that the momentum we want to solve for is equal to the objectβs mass multiplied by its changing position over time. First, letβs solve for that time derivative of position. The time derivative of π as a function of π‘ is equal to π ππ‘ of π‘ squared plus five in the π direction and π‘ cubed plus π‘ in the π direction.

Using the chain rule to differentiate, this is equal to two π‘ in the π direction plus three π‘ squared plus one in the π direction. We can rewrite our expression for π» now to include ππ ππ‘. But we want to solve for π» at a particular value of π‘ when π‘ is equal to what weβve called π‘ sub two or two seconds. So we plug in π‘ sub two for our π‘ values on our equation.

We now have an expression for momentum given entirely in terms of constants or variables whose value we know π and π‘ sub two. Weβre ready to plug in and solve for π». When we do and simplify our expression for our velocity, we find that at this time value itβs equal to four π plus 13 π. Multiplying through by our mass, we find that π» is 28 π plus 91 π. This is the momentum of our mass at π‘ equals two seconds.