CH02-1 Constant Net Force

x-velocity vs. time graph

Theoryfan cart

displacement

Theory

arithmetic mean velocity

Theory

assuming a constant velocity of vx,avg gives the same result.

Fnet vs. time graph

Theoryfan cart

Summary of Analytic Method

Theory

Analytic Method in 1-D

Theory

ExampleA tennis ball has a mass of 0.057 kg. A professional tennis player hits the ball hard enough to give it a speed of 54 m/s (about 121 miles per hour.) The ball moves toward the left, hits a wall and bounces straight back to the right with almost the same speed (54 m/s). As indicated in the diagram below, high-speed photography shows that the ball is crushed about 2 cm at the instant when its speed is momentarily zero, before rebounding. How much time elapses between first hitting the wall and momentarily stopping? What is the net force on the ball during the collision?

video

Numerical (computational) integration

Computation Assumptions: constant net force, small ∆t

Numerical error

Computation Assumptions: constant net force, small ∆t

Error can be decreased by using a smaller time interval. Also, using the mean velocity is clearly a better estimate than using the final velocity.

Decide at this point to accept the error, but select a reasonable time interval to reduce the error.

Error

Summary of Numerical Method

Computation Assumptions: constant net force, small ∆t

At this point, you are using the Euler-Cromer Method (semi-implicit Euler Method) to solve the differential equations:

with

VPython

Write a simulation that models the motion of a fan cart. Clearly state your assumptions and your initial conditions before starting your program.