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 2 years ago
A projectile is launched upward from the ground at 60 m/s.
a. How long will it take to reach its highest point?
b.) How high does it go?
c.) How long does it take to hit the ground?
 2 years ago
A projectile is launched upward from the ground at 60 m/s. a. How long will it take to reach its highest point? b.) How high does it go? c.) How long does it take to hit the ground?

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Yahoo!
 2 years ago
Best ResponseYou've already chosen the best response.0Time = u sin 60 /g H = u^2 sin^2 60 /2g T = 2u sin 60/g

ERoseM
 2 years ago
Best ResponseYou've already chosen the best response.0Thank you but how did you get time and what does the u stand for?

ERoseM
 2 years ago
Best ResponseYou've already chosen the best response.0I'm sorry I don't understand the u?

ERoseM
 2 years ago
Best ResponseYou've already chosen the best response.0Thank you but I'm still confused.

wio
 2 years ago
Best ResponseYou've already chosen the best response.1Okay, we start out with the fundamental idea that everything falls at constant acceleration. We'll call this constant acceleration (with respect to time) \(a\). We'll plug it in for the real value later.

wio
 2 years ago
Best ResponseYou've already chosen the best response.1The velocity is the going to be the anti derivative: \[ v(t) = \int a dt = at+C \]We need to find the constant of integration. Suppose the initial velocity is \(v_0\) (we'll plug it in later). \[ v(0) = v_0 = a(t) +C = C \]So our constant of integration is just the initial velocity \(v_0\). We now have the following equation: \[ v(t) = at + v_0 \]Which is enough to answer question part a

wio
 2 years ago
Best ResponseYou've already chosen the best response.1a. How long will it take to reach its highest point? < Well, the highest point is going to be the maximum position. Remember to find the maximum of a function, you need to find the critical number (when the derivative equals 0). Since velocity is the derivative of position with respect to time, we just need to set it to 0 to find critical numbers: \[ 0 = at + v_0 \implies at = v_0 \implies t = v_0/a \]So the amount of time it will take is just the initial velocity divided by acceleration. Don't worry about that negative sign. Since initial velocity is upward and gravity is downward, the negatives will cancel out. We will get a positive time value.

wio
 2 years ago
Best ResponseYou've already chosen the best response.1b.) How high does it go? < For this we need our position function. Again we find the antiderivative.\[ x(t) = \int v(t)dt = \int at+v_0dt = \frac{1}{2}at^2+v_0t+C \]We have another constant of integration. Again we'll suppose the initial distance of \(x_0\). \[ x(0) = x_0 = \frac{1}2{}a(0)^2 + v_0(0) + C = C \]So once again, our constant of integration is just the initial value. \[ x(t) = \frac{1}{2}at^2+v_0t+x_0 \] We know it will reach it's highest value at \(t = v_0/a\), so let's plug that in: \[ \begin{split} x(v_0/a) &= \frac{1}{2}a(v_0/a)^2+v_0(v_0/a)+x_0 \\ &= \frac{1}{2}\frac{v_0^2}{a}\frac{v_0^2}{a}+x_0 \\ &= \frac{1}{2}\frac{v_0^2}{a}+x_0 \\ &= \frac{v_0^2}{2a}+x_0 \\ \end{split} \] So we have our max distance being: \[ x_{max} = \frac{v_0^2}{2a}+x_0 \]

ERoseM
 2 years ago
Best ResponseYou've already chosen the best response.0Okay, I understand that.

ERoseM
 2 years ago
Best ResponseYou've already chosen the best response.0Now how do I apply real numbers?

ERoseM
 2 years ago
Best ResponseYou've already chosen the best response.0Like where does the sixty actually belong?

wio
 2 years ago
Best ResponseYou've already chosen the best response.1c.) How long does it take to hit the ground? < So we will call the position of the ground \(x_g\). We need to solve for \(t\) given \(x(t) = x_g\)\[ x_g = \frac{1}{2}at^2 + v_0t +x_0 \implies 0 = \frac{1}{2}at^2 + v_0t +x_0 x_g \]This is just the root of a quadratic equation. \[ \begin{split} t &= \frac{(v_0)\pm \sqrt{(v_0)^24(\frac{1}{2}a))(x_0x_g)}}{2(\frac{1}{2}a)} \\ &= \frac{v_0\pm \sqrt{v_0^22a(x_0x_g)}}{a} \end{split} \] This is giving us two times. The earlier time is just when the projectile left the ground, we want the later time, when it hits the ground. Since the \(\sqrt{\ }\) is going to yield a positive value, we set the \(\pm \) to \(+\) to get the later time. \[ t=\frac{v_0 + \sqrt{v_0^22a(x_0x_g)}}{a} \]

wio
 2 years ago
Best ResponseYou've already chosen the best response.1Now we can plug in actual numbers. What we need are \[ a \\ v_0 \\ x_0 \\ x_g \]

wio
 2 years ago
Best ResponseYou've already chosen the best response.1Let's say that upward means 'positive' position, and downward means 'negative' position. \(a\) is gravitational acceleration downward. \(9.8m/s^2\) \(v_0\) is our initial velocity: \(60m/s\) \(x_0\) is our initial position: \(0m\) \(x_g\) is the position of the ground: \(0m\)

wio
 2 years ago
Best ResponseYou've already chosen the best response.1Taking a second look... I think we should try both signs of \(\pm\) rather than just the \(+\), my assumption that that \(+\) version would be higher was incorrect.

wio
 2 years ago
Best ResponseYou've already chosen the best response.1Recap: \[ \begin{array}{rcl} t_{top} &=& v_0/a \\ x_{top} &=& \frac{v_0^2}{2a}+x_0 \\ t_{ground} &=& \max\left(\frac{v_0+ \sqrt{v_0^22a(x_0x_g)}}{a}, \frac{v_0 \sqrt{v_0^22a(x_0x_g)}}{a} \right) \\ a &=& 9.8m/s^2 \\ v_0 &=& 60m/s \\ x_0 &=& 0m \\ x_g &=& 0m \end{array} \]
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