Derivation of the Trajectory of Projectile Motion

[pic 1][pic 2][pic 3]

Introduction

In this report I will try to deliver 4 main sections: (1) Derivation of the trajectory of projectile motion (2) MATLAB source codes (3) Result in MATLAB (4) Discussion about the source codes. Below is the question of this home work.

The trajectory of an object can be modeled as

[pic 4]

where y = height (m),  = initial angle (radians), x = horizontal distance (m), g = gravitational acceleration (=9.81 m/s2), v0 = initial velocity (m/s), and y0 = initial height. Use MATLAB to find the trajectories for y0 = 0and v0 =28 m/s for initial angles ranging from 150 to 750 in increments of 150. Employ a range of horizontal distances from x = 0 to 80 m in increments of 5 m. The results should be assembled in array where the first dimension (rows) corresponds to the different initial angles. Use this matrix to generate a single plot of the heights versus horizontal distances for each of the initial angles. Use this matrix to generate a single plot of the heights versus horizontal distances for each of the initial angles. Employ a legend to distinguish among the different cases, and scale the plot so that the minimum height is zero using the axis command.[pic 5]

1D Motion with Constant Acceleration

In this brief explanation I will define subscripts i as initial, f as final, x and y as the components of vector. We begin with displacement, displacement is defined as change of position, therefore we could write it as:

[pic 6]

Average velocity is particle’s displacement over a time interval during the displacement occurred. It is written as:

[pic 7]

Also, we may define an instantaneous velocity as:

[pic 8]

For a trajectory motion (explained later) we assume the acceleration is always constant, thus we may define constant acceleration by assuming initial time is zero and final time is t as:

[pic 9]

By reforming the formula above, we can get the velocity of particle at any time which is very useful.

[pic 10]

Velocity with constant acceleration varies linearly in time as you can see from the formula above. Therefore, we can get the average velocity as:

[pic 11]

Now, we can substitute equation (6) to equation (2), therefore we will get:

[pic 12]

Next, by substituting equation (5) to (7), we will get:

[pic 13]

2D Motion with Constant Acceleration

In two dimension, we can write position and velocity vector as:

[pic 14]

[pic 15]

Because of constant acceleration a, its components are also constant. Therefore, we can rewrite equation (10) for 2D as: