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Numerical integration of projectile motion

In the last workshop, you did a little numerical differentiation: using known positions to figure out velocity and acceleration as a function of time.

Today, we turn it around: you will use known velocity and acceleration to figure out -- and graph -- position of a ball in motion. This is a simple example of numerical integration.

Here are the starting conditions:

a ball starts at the origin, px = 0 and py = 0 at time t = 0

its initial velocity is v = 40 m/s at an angle θ = 60 degrees above the horizontal

Your job is to make an approximate model of the ball's trajectory. Using a piece of good graph paper, draw a set of axes, running from 0 to 160 meters horizontally, and 0 to 120 meters vertically. Then

assume that the ball moves with constant velocity for 1-second intervals: from t=0 to t=1, assume it moves in straight line with the initial speed

assume that at the end of each second, the ball's velocity changes all of a sudden to its new value for the next second

and then follow the motion for another full second with that new, constant velocity

You may pretend for simplicity that this experiment takes place deep underground, in a mineshaft, where the local acceleration due to gravity is g = 10 m/s^2 downwards. That will simplify your calculations.

This will be pretty easy if you make a table showing the components of position, velocity and acceleration over time. I'll start you off ...

Time px py vx vy ax ay (m) (m) (m/s) (m/s) (m/s^2) (m/s^2) --------------------------------------------------------------- 0 0 0 20 35 0 -10 1 __ __ ___ ___ __ ___ 2 3 4 5 6 ---------------------------------------------------------------

show the ball's motion from t=0 to t=6 seconds

after you have drawn the path, label each velocity vector as V1, V2, ..., V6

how much does the velocity change from one second to the next? Show this on your graph by drawing the vector equation V2 - V1. Use a dashed line for the negative copy of V1, and use a different colored pen or pencil to draw the resultant vector (which will be the acceleration)

Adapted from Prof. Michael Richmond. Sorry, we could not find this page | RIT CIS - Center for Imaging Science

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Numerical integration of projectile motion

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