Commit 1c9ce7ea authored by bentelk's avatar bentelk
Browse files

Typo corrections and minor updates in the explanations

parent 865fd726
......@@ -8,20 +8,30 @@
"## Animate a bouncing ball\n",
"\n",
"Solve the equations of motion for the position and velocity of a ball as a function of time. \n",
"Change in position $x$ and velocity $v$ over the time interval $\\Delta t$ are\n",
"Changes in position $x$, $y$ and velocity $v$ over the time interval $\\Delta t$ are\n",
"$$\n",
"\\Delta x = v \\cdot \\Delta t \\\\\n",
"\\Delta v = a \\cdot \\Delta t\n",
"\\Delta x = v_x \\cdot \\Delta t \\\\\n",
"\\Delta y = v_y \\cdot \\Delta t \\\\\n",
"\\Delta v_y = a \\cdot \\Delta t\n",
"$$\n",
"\n",
"With only gravitational acceleration acting we have $a = g = 9.81 \\text{m}/\\text{s}^2$\n",
"Assume $v_x$ is constant and only gravitational acceleration is acting, i.e. $a = g = 9.81 \\text{m}/\\text{s}^2$. If position $y < 0$ the ball bounces with a coefficient of $coe_b = 0.7$.\n",
"\n",
"\n",
"Let $x_1$ and $v_1$ be the initial position and velocity and $x_2$ and $v_2$ denote the position and velocity after the time interval $Δt$. We can use the above equations, and $Δx=x_2−x_1$, $Δv=v_2−v_1$, to solve for $x_2$ and $v_2$.\n",
"Let $x_1$, $y_1$ and $v_{x1}$, $v_{y1}$ be the initial position and velocity and $x_2$, $y_2$ and $v_{x2}$, $v_{y2}$ denote the position and velocity after the time interval $Δt$. With the above equations and $Δx=x_2−x_1$, $Δy=y_2−y_1$, $Δv=v_2−v_1$, $v_x = constant$, solve for $x_2$, $y_2$ and $v_2$ (dropping the index $y$ from now on, i.e. $v_y = v$, as $v_x = v_{x0}$ = const).\n",
"$$\n",
"x_2=x_1+v_1Δt \\\\\n",
"x_2=x_1+v_{x0}Δt \\\\\n",
"y_2=y_1+v_1Δt \\\\\n",
"v_2=v_1+aΔt\n",
"$$ "
"$$ \n",
"if $y < 0$ the ball bounces: \n",
"$$\n",
"y_i = 0\\\\\n",
"v_i = -v_{i-1} * coe_b\n",
"$$\n",
"\n",
"\n",
"<b> *Note:* </b> Be patient as the animation takes a few seconds to display!"
]
},
{
Supports Markdown
0% or .
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!
Please register or to comment