Question Video: Finding the Momentum of a Body at a Given Time given an Expression of Its Position with Time

Video Transcript

A body of mass seven kilograms is moving in a straight line. Its position vector at a time 𝑡 is given by the relation 𝑟 as a function of 𝑡 equals 𝑡 squared plus five quantity in the 𝑖 direction plus 𝑡 cubed plus 𝑡 quantity in the 𝑗 direction, where the magnitude of 𝑟 is measured in metres and 𝑡 in seconds. Determine the body’s momentum after two seconds.

We can record the body’s mass of seven kilograms as 𝑚. The time value we’re interested in — two seconds — we’ll call 𝑡 sub two. We’re given the position of our object as a function of time — 𝑟 as a function of 𝑡 — and based on that want to solve for its momentum, which we’ll represent by the letter capital 𝐻.

To begin on our solution, let’s recall the mathematical relationship describing momentum. The momentum of an object with mass is equal to that object’s mass times its velocity. We know that an object’s velocity is defined as its change in position over time — 𝑑𝑟 𝑑𝑡.

Combining these relationships, we can write that the momentum we want to solve for is equal to the object’s mass multiplied by its changing position over time. First, let’s solve for that time derivative of position. The time derivative of 𝑟 as a function of 𝑡 is equal to 𝑑 𝑑𝑡 of 𝑡 squared plus five in the 𝑖 direction and 𝑡 cubed plus 𝑡 in the 𝑗 direction.

Using the chain rule to differentiate, this is equal to two 𝑡 in the 𝑖 direction plus three 𝑡 squared plus one in the 𝑗 direction. We can rewrite our expression for 𝐻 now to include 𝑑𝑟 𝑑𝑡. But we want to solve for 𝐻 at a particular value of 𝑡 when 𝑡 is equal to what we’ve called 𝑡 sub two or two seconds. So we plug in 𝑡 sub two for our 𝑡 values on our equation.

We now have an expression for momentum given entirely in terms of constants or variables whose value we know 𝑚 and 𝑡 sub two. We’re ready to plug in and solve for 𝐻. When we do and simplify our expression for our velocity, we find that at this time value it’s equal to four 𝑖 plus 13 𝑗. Multiplying through by our mass, we find that 𝐻 is 28 𝑖 plus 91 𝑗. This is the momentum of our mass at 𝑡 equals two seconds.