The position of a particle as a function of time is given by 𝓇 = ( 5.0î ＋4.0ĵ )t² m where t is in seconds. c. What is the particle's speed at t = 0, 2, and 5 s?

Video transcript

Hey, everyone, this problem is dealing with vector notation. So let's see what they are asking us and what they're giving us. So they tell us first that acceleration is the rate of change in velocity over time. And then they give us an expression for a position given as a function of time right here. And they're asking us to determine the particle's velocity. So this is a little bit tricky because they tell us that acceleration is the rate of change in velocity over time. But they don't go as far as to say that velocity is given as the rate of change in position over time. But if we can recall that, then we know what we need to do to find the velocity because we are given that particles position. So we'll write that as V of T equals D R D T where R equals five I plus four J times T squared. And we're working in meters. So we're going to take the derivative of that to find our velocity. So that looks like five I plus four J T squared D T, that'll be two Times five I plus four J T in our units. Now, after we have our derivative is meters per second, we will multiply through that too and come up with 10 I plus for J sorry plus eight J, we apply That two to each of the terms eight J T. And so the magnitude is given by the square root of the sum of squares. And that's gonna be 10 T squared plus 80 squared. And we come out of that with a V of tea Equals 12.8 times Tm/s. And so what they're asking us is to determine the particle's velocity at time equals zero in the time equals two seconds. So V of zero will just plug that in for tea and that goes to zero, zero m/s is the answer for part one And v of two seconds Is 12.8 times to plug that into our calculators and get 25.6 meters per second. So that is the answer to this problem. And if we look at our choices that aligns with cancer de, that's all we have for this one, we'll see you in the next video.