1. ## Help with ODE

I need some help with this homework problem:

1. A two inch diameter craft ball is thrown vertically. This initial velocity of the ball is 20 ft/s.
1. Neglecting drag, draw a free body diagram and formulate a first order ODE that governs the velocity of the ball. Is the ODE linear or non-linear?
My answer: Linear, F=ma so dv/dt * m =m*g hence, dv/dt = g.

2. Use Euler’s first order method to determine the “drag free” time required to achieve maximum elevation. Check your solution with the analytic solution to the ODE that governs the motion of the ball.
My answer: I do not know if my ODE in part 1 is correct?
The analytic solution will be V = g*t +Vo.

3. The terminal velocity of the ball (during a free fall) is determined to be 20 ft/s, and the drag force is known to be proportional to the velocity squared. Draw a free body diagram and formulate a first order ODE that governs the velocity of the ball. Is the ODE linear or non-linear?
My answer: Non-linear. F = mg - k*v^2 therefore... dv/dt = g -(k*v^2)/m

4. Use Euler’s method to estimate the “actual” time to achieve maximum elevation.
My answer: N/A ... How do I derive the ODE for this part?

Any help is greatly appreciated! I just don't quite understand how to derive the correct ODE's.

2. Originally Posted by The0wn4g3
I need some help with this homework problem:

1. A two inch diameter craft ball is thrown vertically. This initial velocity of the ball is 20 ft/s.
1. Neglecting drag, draw a free body diagram and formulate a first order ODE that governs the velocity of the ball. Is the ODE linear or non-linear?
My answer: Linear, F=ma so dv/dt * m =m*g hence, dv/dt = g.

2. Use Euler’s first order method to determine the “drag free” time required to achieve maximum elevation. Check your solution with the analytic solution to the ODE that governs the motion of the ball.
My answer: I do not know if my ODE in part 1 is correct?
The analytic solution will be V = g*t +Vo.
If $x$ is positive upwards then the gravitational force is negative, and the ODE is:

$\frac{d^2x}{dt^2}=-g \approx -9.81 \text{ m/s}^2$

or for velocity (with positive upwards):

$\frac{dv}{dt}=-g \approx -9.81 \text{ m/s}^2$

CB