Mini project: Projectile motion
Problem formulation
A small material particle of mass m kg is thrown from the height of H meters above the ground level towards a fence of height h m located l meters from the projectile initial location. The magnitude of the initial velocity (speed) is V m/sec. Find the angle α with the horizontal at which the projectile should be thrown in order to just clear the top of the fence. Plot a trajectory of the projectile. If there are more than one trajectory satisfying the conditions of the problem, plot all of them in one figure. Assume that the effect of air resistance is negligible. Use the gravitational acceleration g=9.81 m/(sec^(2).
Part 1
Solve the problem interactively, step by step with specific parameters H = 2; h = 3.5; l = 20; V = 15.
Step 1: Solving the equations of projectile motion
Use Newton's Second Law to set up the ODEs for the horizontal and vertical components of position of the projectile. Find the genearal solutions to the ODEs. Then determine the four arbitrary constants in this general solution using the initial conditions for position and velocity.
The solution involves unknown direction of the initial velocity.
Step 2: Finding the time when projectile reaches the fence
Use x(t) to find the time tfence when the projectile reaches the fence location, that is, when
.
Step 3: Setting up the equation for and finding
The height must be equal to the height of the fence: