CALC A 100 g bead slides along a frictionless wire with the parabolic shape y = (2m⁻¹) x^2. a. Find an expression for aᵧ, the vertical component of acceleration, in terms of x, vₓ, and aₓ. Hint: Use the basic definitions of velocity and acceleration.

Video transcript

Hey, everyone. So today we're dealing with the problem regarding derivatives and relations between position acceleration and velocity. So we're being told that the projectiles parabolic trajectory, the projectile with a mass of 150 g is given by the equation Y is equal to four X squared. So using standard definitions of velocity acceleration are being asked to find an expression that expresses a Y, the accelerations vertical component in terms of X V X and A X where V X horizontal velocity and A X is horizontal acceleration. So let's start with what we're given. We're told that that why is equal to four X squared? Now this is the equation of our Parabola. So let's start with that. So if we differentiate this with respect to time, because this is essentially a position graph, right? Any position of X tells us where we are in Hawaii, so any horizontal position will compute to a vertical position. So we take the derivative this well with respect to time. However, we make it a sort of position versus time graph if we derive it as such, which will look like this. If we take the, the derivation of why? And with respect to T, well, it'll be the D T using our rules. We'll get that. This is ah two and 2, four x so on, Which gives us eight x multiplied by DX DT. And in terms of velocity, where we could say Dy over DT is equal to the vertical velocity. And we could say that the X over D T is equal to the horizontal velocity. Then this becomes that the vertical velocity Is equal to eight x V X assuming this to be true with these definitions. So from here we have our uh vertical velocity. So to find an expression in terms of acceleration, we need to differentiate this yet again with respected time. Let's use blue. So the horizontal velocity due to time will be the will be the horizontal acceleration and the vertical velocity with respect to time will be the vertical acceleration. So deriving this, we get D Y over D T is equal to T over T T eight x fi x expanding this and simplifying we get, Hey, one is equal to eight x D V X over D T plus V X D eight over x over D T. Thanks to our calculus rules which simplifies to a final answer of A Y is equal to eight X A X Plus eight VX Squared. Therefore, the accelerations vertical component in Y is represented with the equation eight X A X plus eight V X squared or answer choice. See, I hope this helps and I look forward to seeing you all in the next one.