P01R2021 – Physics Rocks

Physics Rocks

A first rock is thrown from ground level with an upward speed $v_0=20\ m/s$ while a second rock is dropped from rest at a height $H_2=100\ m$ above the ground. You may approximate $g\approx 10\ m/s^2$ for your calculations. Show your work and include units for all numerical answer.

1. Write the kinematic equations $y_1\left(t\right)$ and $v_1\left(t\right)$ for the first rock.

\boxed{
\left\{ \begin{array}{c}
\displaystyle{y_1\left(t\right)=-\frac{1}{2}gt^2+v_0t} \\
\\
v_1\left(t\right)=-gt+v_0 \ \ \ \ \ \end{array}
\right.
}

2. Write the kinematic equations $y_2\left(t\right)$ and $v_2\left(t\right)$ for the second rock.

\boxed{
\left\{ \begin{array}{c}
\displaystyle{y_2\left(t\right)=-\frac{1}{2}gt^2+H_2} \\
\\
v_2\left(t\right)=-gt \ \ \ \ \ \ \ \ \ \ \ \ \ \ \end{array}
\right.
}

3. At what time are the rocks at the same height?

The rocks are at the same height at time $t_f$ such that $y_1\left(t_f\right)=y_2\left(t_f\right)$ and we solve for $t_f$ as follows

\begin{aligned}
y_1\left(t_f\right)=y_2\left(t_f\right)\ \ \ \ &\Rightarrow \ \ \ \ -\frac{1}{2}gt^2_f+v_0t_f=-\frac{1}{2}gt^2_f+H_2 \\
\\
&\Rightarrow \ \ \ \ \ t_f=\frac{H_2}{v_0}
\end{aligned}

The rocks are at the same height at time $t_f$ equal to

\boxed{t_f=\frac{H_2}{v_0}=\frac{100}{20}=5\ s}

4. When the rocks are at the same height, what is the velocity of the first rock? Is it moving upward or downward?

\boxed{v_1\left(t_f\right)=-gt_f+v_0=-10\cdot 5+20=-30\ m/s}

5. When the rocks are at the same height, what is the velocity of the second rock?

\boxed{v_2\left(t_f\right)=-gt_f=-10\cdot 5=-50\ m/s}