How Do You Correctly Model Projectile Motion with Drag?

[Freyster98]

Homework Statement

If we include a crude model for the drag force in which the net acceleration on the ball kicked by a football player is given by:
a = (-k*v_x)i + (-g-k*v_y)j.
Derive the equations describing the horizontal and vertical positions as functions of time.
k=.031 (1/s), v_o=69 (ft/s), \Theta₀=45.

Homework Equations

The Attempt at a Solution

I solved for v_x and v_y using the information given (v_x=v₀cos\Theta, v_y=v₀sin\Theta ) plugged these values, along with k, into the acceleration equation. I took the integral of both the horizontal and vertical components independently to get velocity, then integrated that to get the position. The problem is, the horizontal velocity comes out to be -1.51*t, and horizontal position is -.755*t^2, which is obviously wrong because it would be moving backwards the instant it is kicked. What am I doing wrong here? Should I not be solving for v_x and v_y, and leaving those as variables as well?

 

[tiny-tim]

… What am I doing wrong here? Should I not be solving for v_x and v_y, and leaving those as variables as well?

Hi Freyster98!

erm … yes …

v_x and v_y are definitely variables …

the original equation says that the acceleration is -g vertically, and -k (the drag coefficient) times the instantaneous velocity horizontally.

 

[musichael]

Im working on the same problem. Do I plug in the initial values and integrate. or do i leave the initial velocity as v0 and then integrate?

 

[tiny-tim]

Im working on the same problem. Do I plug in the initial values and integrate. or do i leave the initial velocity as v0 and then integrate?

Hi musichael!

Leave v₀ until the end …

v_x and v_y are variables …

integrate, and you will get a constant …

at that stage you use v₀ to find what the constant is.

 

[CEL]

The horizontal velocity is v_x = v_0 cos \theta_0 + a_x t, where a_x = -k v_x is a variable and not a constant as you used.