Question

[Computer] Use suitable graph-plotting software to plot graphs of the trajectory (2.36) of a projectile thrown at 45 ^above the horizontal and subject to linear air resistance for four different values of the drag coefficient, ranging from a significant amount of drag down to no drag at all. Put all four trajectories on the same plot. [Hint: In the absence of any given numbers, you may as well choose convenient values. For example, why not take \(v_{x \mathrm{o}}=v_{y \mathrm{o}}=1\) and \(g=1 .\) (This amounts to choosing your units of length and time so that these parameters have the value \(1 .\).) With these choices, the strength of the drag is given by the one parameter \(v_{\text {ter }}=\tau,\) and you might choose to plot the trajectories for \(v_{\text {ter }}=0.3,1,3,\) and \(\infty\) (that is, no drag at all), and for times from \(t=0\) to \(3 .\) For the case that \(v_{\text {ter }}=\infty\) you'll probably want to write out the trajectory separately.]

Short answer

Use a graph plotter to visualize projectile paths for different drag coefficients, comparing trajectories from no drag to significant drag.

Step-by-step solution

Understand the Trajectory Equation

Start by identifying the trajectory equation for a projectile subject to linear air resistance. The general form involves integrating the motion equations with a drag force term. The air resistance force is proportional to the velocity with a drag coefficient involved.

Analyze Given Values and Conditions

Based on the problem statement, we are given initial velocities both x and y such that \(v_{x,o} = v_{y,o} = 1\). The gravitational acceleration \(g = 1\) is also set. The variable \(v_{ter}\), which refers to terminal velocity, will be used for four cases: \(v_{ter} = 0.3, 1, 3, \text{and } \infty\).

Recognize the Effects of Drag Coefficient

The drag coefficient affects the projectile's trajectory. Lower terminal velocities mean higher drag and vice versa. Four cases of \(v_{ter}\) will depict different levels of drag: significant drag \(v_{ter} = 0.3\), moderate drag \(v_{ter} = 1\), low drag \(v_{ter} = 3\), and no drag \(v_{ter} = \infty\). A trajectory equation needs to be plotted for each of these values.

Plot with Graph-Plotting Software

Using a graph-plotting tool like MATLAB, Python with Matplotlib, or Desmos, plot the trajectory equations. For each case, compute the trajectory path from \(t = 0\) to \(t = 3\) by solving the equations of motion numerically if needed. Ensure to label each trajectory with its corresponding \(v_{ter}\) and drag conditions.

Interpret the Trajectories

Analyze and compare the plotted trajectories to infer how the drag affects the motion. The case with no drag (\(v_{ter} = \infty\)) will provide the longest trajectory, while significant drag (\(v_{ter} = 0.3\)) will show a sharp drop and shorter path.

Key concepts

Linear Air Resistance

When dealing with projectile motion, it's important to consider the effects of air resistance. Linear air resistance is a simplification where the force of the air resisting the object's motion is proportional to its velocity. This means that as a projectile moves faster, the air resistance, or drag force, increases, directly proportional to that speed.

In equations of motion, this force is expressed as a negative term added to the velocity. This term complicates the trajectory path, making it more realistic compared to idealized scenarios like vacuum conditions. If not considered, a projectile might appear to travel further than it actually would in real life, especially in conditions with significant air drag.

Your trajectory equation will include this drag term, altering both the horizontal and vertical components of motion, which dictates the arc and reach of your projectile.

Drag Coefficient

The drag coefficient, often denoted by a symbol like \(C_d\), is a crucial factor in aerodynamics and physics, indicating how much drag force will affect a moving object. It's determined by the shape of the object, its surface texture, and the density of the air through which it travels.

For our projectile motion problem, the terminal velocity \(v_{ter}\) is influenced by the drag coefficient. Terminal velocity is the speed at which the force of gravity is balanced by the drag force, resulting in no net acceleration.

Four scenarios depict different drag conditions using \(v_{ter}\):

• High drag (\(v_{ter} = 0.3\)) limits the projectile significantly, creating a steep drop.
• Moderate drag (\(v_{ter} = 1\)) provides a balanced curve.
• Low drag (\(v_{ter} = 3\)) allows for a longer flight.
• No drag (\(v_{ter} = \infty\)) simulates an ideal, unimpeded motion, reaching the furthest distance.

Choosing appropriate values for \(v_{ter}\) lets us see and analyze the varying effects of drag on the projectile's path.

Graph Plotting Software

Graph plotting software is a powerful tool for visualizing projectile trajectories under different conditions. Tools like MATLAB, Python's Matplotlib, or Desmos are ideal because they allow us to input complex mathematical equations and visualize their outcomes effortlessly.

To start, you set your initial conditions and parameters, such as initial velocities and drag coefficients. Then, by solving the equations of motion for each scenario using the software, you get a visual representation of each trajectory over time, from \(t = 0\) to \(t = 3\).

The advantage of using such software lies in its ability to handle numerical calculations accurately and present data that can be quickly interpreted. It helps in identifying the influence of drag, showing accurately how each increase in drag affects the projectile's motion.

With labeled axes, legends, and color-coded trajectories, it's easy to compare each path and understand the real-world implications of air resistance in projectile motions.