Parabolic motion An arrow is shot into the air and moves alon...

Question

Parabolic motion An arrow is shot into the air and moves along the parabolic path y=x(50−x) (see figure). The horizontal component of velocity is always 30 ft/s. What is the vertical component of velocity when (a) x=10 and (b) x=40?

Accepted Answer

Welcome back, everyone. A projectile is launched and its path can be modeled by the equation Y equals 3x multiplied by 40 minus X, where Y represents the height and feet and X is the horizontal distance in feet. Given that the horizontal component of velocity is 25 ft per second, what is the vertical component of velocity when X equals 10? So essentially what we're going to do is use the position function and we're going to differentiate it to understand which one is the correct answer. A says 3000 ft per second, B 250 ft per second, C 1500 ft per second, and D 750 ft per second. So we're given Y equals 3 X multiplied by 40 minus X, right? So we have a relationship between the vertical distance and the horizontal distance. This is how we can interpret it. And now we can distribute the terms we can say that Y equals 120 x minus 3 x 2, and we have to understand that Y is a function of X and it is also a function of time. So relative to time, Y and X are both changing, right? So when we differentiate the Y divided by DT, we have to apply the chain rule. So this is going to be the Y divided by DX we're differentiating with respect to X, and then based on the chain rule, we have to multiply by D X divided by DT, the derivative of X relative to time. Let's go ahead and do that. So our D Y divided by DT, the rate of change of Y with respect to time, is equal to 120, right? Because the derivative of X is 1. And because X is a function of time, we're multiplying by D X divided by D T, and then we are subtracting the derivative of 3 x 2, which is 6 X based on the power rule multiplied by the X divided by DT, right, because once again Xs as a function of time, and now we can simplify it. We can essentially factor out 6 the X divided by the T. And then in parentis, well, we are going to have 20. Minus X. So we have our expression. Let's understand that on the left hand side, that's the vertical component of velocity because we have a rate of change of height. So that's vertical component of velocity. And for the X divided by the T, that's the horizontal one. So all that we have to do is simply use the given data and substitute. So DY. Divided by D T. When X is equal to 0, so we're going to say that X is equal to, I'm sorry, X is equals 10, right? That's what we're given. So X is equal to 10. This must be equal to 6 multiplied by now D X divided by DT we're given the horizontal component of velocity of 25, so 25. Feet per second we can also include it. In our equation, we can say that the X divided by D T is equal to 25, and we're multiplying that by 20. Minus X. So essentially 20 minus our X is 10. So we get 6 multiplied by 25, which is 150 multiplied by 10. And this gives us. 1500 ft. Per second which corresponds to the answer choice C. Thank you for watching.

Parabolic motion An arrow is shot into the air and moves along the parabolic path y=x(50−x) (see figure). The horizontal component of velocity is always 30 ft/s. What is the vertical component of velocity when (a) x=10 and (b) x=40? <IMAGE>

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Parabolic ​motion An arrow is shot into the air and moves along the parabolic path y=x(50−x) (see figure). The horizontal component of velocity is always 30 ft/s. What is the vertical component of velocity when (a) x=10 and (b) x=40? <IMAGE>

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Parabolic motion An arrow is shot into the air and moves along the parabolic path y=x(50−x) (see figure). The horizontal component of velocity is always 30 ft/s. What is the vertical component of velocity when (a) x=10 and (b) x=40? <IMAGE>