Effect of air Resistance
In the real world, air resistance has a marked effect on the motion of a projectile. If the object is light, then it does not have the inertia to push through the air. Air resistance, often called drag creates an additional force on the projectile that acts in the opposite direction to the velocity.
For objects that move through air and at high velocity, the air resistance is proportional to the square of the velocity.
To calculate the path of the projectile with air resistance, we need to know the forces acting on the projectile. We know that gravity will always act vertically downward on the projectile. The air resistance creates a force, that is always in the opposite direction to the velocity.
From Newton’s second law, F = m.a, we can see that this force will have its effect by changing the acceleration of the cannon ball. In the x direction it will act to decelerate the projectile reducing its range. It will also act on the vertical component of velocity slowing the projectile on both the upward and downward part of the motion.
The motion is described by summing the forces in the x and y directions. In the x-direction, To find the path of the projectile we must solve two differential equations.
Since acceleration is the second derivative of the position we can write a differential equation which the motion.
These are non-linear and so cannot be solved analytically. Numerical solution is possible however. To solve numerically, we make the assumption that the over a small enough interval of time we can come approximate the solution by applying the linear equations of motion, to obtain velocities and then the position of the projectile.
Before we do that however we should rewrite the differential equations. We know that the drag is proportional v2. But v is a vector so, v2 = v v = v vx and v vy
The constant term in is made up of several factors such as the density of air, cross-sectional area, drag coefficient, but none of these vary with time for our purposes, so we are okay to bundle them up as a number. Let’s call it k, it is actually,
We can compute the new velocities in the x and y directions after a short time dt. The new velocities will be. Once we have the new velocities we can calculate the new position of the cannonball..
~ END OF VOLUME-I ~