"then at time t$$_{i+1}$$=t$$_i$$+Δt the position and velocity of the particle can be approximated by\n",
"then at time $t_{i+1}=t_i+Δt$ the position and velocity of the particle can be approximated by\n",
"\n",
"$$\\vec{x}_{i+1} = \\vec{x}_i+Δt\\vec{v}_i$$\n",
"\n",
"$$\\vec{v}_{i+1} = \\vec{v}_i+Δt\\vec{a}_i$$\n",
"\n",
"So if we start at time t$_0$=0 at position $\\vec{x}_0$ and velocity $\\vec{v}_0$, then provided we can calculate the force on the particle, and hence the acceleration $\\vec{a}_i$, and use small enough time steps Δt, we can iteratively trace out the path by repeated use of the above relations.\n",
"So if we start at time $t_0=0$ at position $\\vec{x}_0$ and velocity $\\vec{v}_0$, then provided we can calculate the force on the particle, and hence the acceleration $\\vec{a}_i$, and use small enough time steps Δt, we can iteratively trace out the path by repeated use of the above relations.\n",
"\n",
"### Forces on Particle\n",
"\n",
...
...
%% Cell type:markdown id:abaa6e4a tags:
## Checkpoint 5
### Aim
To write two Python programs to numerically analyse the path of a projectile under gravity and drag force. There are two tasks, one to display the trajectory and the second to display the relation between final kinetic energy and launch angle.
### Numerical Integration
In this checkpoint we will use the simplest numerical integration technique called First Order Euler. This involves iteratively calculating the position and velocity at a set of times t$_i$=iΔt for i=0,1,2,… where we assume that the position and velocity at time t$_{i+1}$ depend only on the position and velocity at time t$_i$. In particular if at time t$_i$ the particle has
$$\text{Position} = \vec{x}_i$$
$$\text{Velocity} = \vec{v}_i$$
$$\text{Acceleration} = \vec{a}_i$$
then at time t$$_{i+1}$$=t$$_i$$+Δt the position and velocity of the particle can be approximated by
then at time $t_{i+1}=t_i+Δt$ the position and velocity of the particle can be approximated by
$$\vec{x}_{i+1} = \vec{x}_i+Δt\vec{v}_i$$
$$\vec{v}_{i+1} = \vec{v}_i+Δt\vec{a}_i$$
So if we start at time t$_0$=0 at position $\vec{x}_0$ and velocity $\vec{v}_0$, then provided we can calculate the force on the particle, and hence the acceleration $\vec{a}_i$, and use small enough time steps Δt, we can iteratively trace out the path by repeated use of the above relations.
So if we start at time $t_0=0$ at position $\vec{x}_0$ and velocity $\vec{v}_0$, then provided we can calculate the force on the particle, and hence the acceleration $\vec{a}_i$, and use small enough time steps Δt, we can iteratively trace out the path by repeated use of the above relations.
### Forces on Particle
If we consider a particle of mass m travelling with velocity $\vec{v}$ subject to gravity and drag from a fluid as shown below, then the drag force is given by
$$ \vec{F}_d=−\frac{1}{2}\rho AC_Dv^2\hat{v}$$
where $\rho$ is the density of the fluid, A is the cross-sectional area of the particle, C$_D$ is the drag coefficient and $\hat{v}=\vec{v}/|v|$, being the unit vector in the direction of the velocity.
Thus the acceleration of of the particle is given by $\vec{a}=\frac{1}{m}\vec{F}_d−g\hat{j}$ where g is acceleration due to gravity, being 9.81ms$^{−2}$. We can write this as:
$$ \vec{a} =−\beta v^2\hat{v}−g\hat{j}$$
where $\beta=\frac{\rho AC_D}{2m}$, where $\beta$ is the normalised drag coefficient for the system.
In this problem the position, velocity and acceleration are all two dimensional vectors so they have a horizontal and vertical component that you have to treat separately in the calculation.
So in terms of components in x (horizontal) and y (vertical) we can write:
where the components of acceleration from the above equations are:
$a_x=−\beta vv_x$ and $a_y=−\beta vv_y−g$
where v is the magnitude of the velocity, so being $v=\sqrt{v^2_x+v^2_y}$
### Task
**Task 1:** Write a Python program to plot out the trajectory of a projectile starting at position $\vec{x}_0=(0,0)$ for specified initial velocity and normalised drag coefficient. You program should prompt for
- Magnitude of the initial velocity v$_0$ in ms$^{−1}$.
- Angle θ from the horizontal of initial velocity in degrees.
- β the normalised drag coefficient.
- Δt the step interval in seconds.
The path should be traced and displayed using Matplotlib from the inital starting position until the particle re-crosses the x-axis. Your graph should have a suitable title and axis labels. Your program should also print the range of the projectile to the terminal.
**Task 2:** Modify the above program to produce a plot of the ratio of final kinetic energy to initial kinetic energy against lauch angle θ in the range 0$\to$90$^{\circ}$. Your program should prompt for :
- Initial velocity v$_0$.
- Normalised drag coefficient $\beta$.
- Timestep $\Delta$t.
and plot out, using Matplotlib, a graph of K$_f$/K$_i$ against $\theta$.
### Checkpoint
- To help you with this checkpoint we've already provided you with a program skeletong structure below.
- Test your program with the following conditions:
