Question

help!

Answer 1

What excatly is your problem?

Answer 2

As \(y\) components we've got the gravity, which faces downwarts and our toss, which faces upwards:

1) Highest point: For this part we use the fact, that at the highest point the velocity is zero:
\[
v_y = v_{0y} - gt = 0 \Rightarrow t = \frac{v_{0y}}{g}
\]

2) Maximum height: We've just calculated the time it needs to get to the highest point. Therefore we can plug this time into the formula of position with respect to time:\[
r(t) = \underbrace{r_0}_{72ft} + v_{0y}t - \frac{1}{2}gt^2 \Rightarrow r = r(t) = \underbrace{r_0}_{72ft} + v_{0y} \frac{v_{0y}}{g} - \frac{1}{2}g\left( \frac{v_{0y}}{g}\right)^2
\]\[
= \underbrace{r_0}_{72ft} + \frac{v_{0y}^2}{g} - \frac{1}{2}\frac{v_{0y}^2}{g}\]

Answer 3

\[t_{up} = \frac{v_{0y}}{g}\]
3) Total time of travel: To solve this we can use two things.
- We know the time it takes for the stone to get to it's highes point.
- We assume the beach to be at a height of \(h = 0ft\)
The stone falling down can be seen as a free fall from a given height \(h_{max}\), we calculated in section 2. The equation for free fall is at follows:
\[
r(t_{end}) = h_{max} - \frac{1}{2}gt^2 = \underbrace{\left[ r_0 + \frac{v_{0y}^2}{g} - \frac{1}{2} \frac{v_{0y}^2}{g} \right]}_{h_{max}}- \frac{1}{2}gt^2
\]
If we now solve for \(t\) which should give us the time it takes to fall from \(h_{max}\) down to the beach:
\[
t = \sqrt{\frac{2h_{max}}{g}} = \sqrt{\frac{2\left[ r_0 + \frac{v_{0y}^2}{g} - \frac{1}{2} \frac{v_{0y}^2}{g} \right]}{g}}
\]
The total time of travel can now be calculated as:
\[
t_{total} = t_{up} + t_{down} = \frac{v_{0y}}{g} + \sqrt{\frac{2\left[ r_0 + \frac{v_{0y}^2}{g} - \frac{1}{2} \frac{v_{0y}^2}{g} \right]}{g}}
\]

Answer 4

4) Velocity at the impact: The impact velocity can now easily be calculated by the time of travelling downwards and the acceleration due to gravity:
\[
v_{impact} = - gt_{down} = -g \sqrt{\frac{2\left[ r_0 + \frac{v_{0y}^2}{g} - \frac{1}{2} \frac{v_{0y}^2}{g} \right]}{g}}
\]

Answer 5

I think the formulas should be right that way, but please try to make yourself clear what i did and the physical meaning of each step. If anything is unclear or you find a mistake please don't hesitate to aks.