How Do You Correctly Model Projectile Motion with Drag?

In summary: This results in v_x = v_0 cos \theta_0 / (1 + k t). The vertical velocity is v_y = v_0 sin \theta_0 + a_y t, where a_y = -g - k v_y is a variable and not a constant as you used. This results in v_y = (v_0 sin \theta_0 + g) / (1 + k t). The horizontal position is x = x_0 + \int v_x dt, where the integral is from 0 to t. The vertical position is y = y_0 + \int v_y dt, where the integral is from 0 to t.In summary, the conversation discusses deriving equations for the
  • #1
Freyster98
49
0

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*vx)i + (-g-k*vy)j.
Derive the equations describing the horizontal and vertical positions as functions of time.
k=.031 (1/s), vo=69 (ft/s), [tex]\Theta[/tex]0=45.

Homework Equations


The Attempt at a Solution



I solved for vx and vy using the information given (vx=v0cos[tex]\Theta[/tex], vy=v0sin[tex]\Theta[/tex] ) 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 vx and vy, and leaving those as variables as well?
 
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  • #2
Freyster98 said:
… What am I doing wrong here? Should I not be solving for vx and vy, and leaving those as variables as well?

Hi Freyster98! :smile:

erm :redface: … yes …

vx and vy are definitely variables …

the original equation says that the acceleration is -g vertically, and -k (the drag coefficient) times the instantaneous velocity horizontally.
 
  • #3
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?
 
  • #4
musichael said:
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! :smile:

Leave v0 until the end

vx and vy are variables …

integrate, and you will get a constant …

at that stage you use v0 to find what the constant is. :wink:
 
  • #5
The horizontal velocity is [tex]v_x = v_0 cos \theta_0 + a_x t[/tex], where [tex]a_x = -k v_x[/tex] is a variable and not a constant as you used.
 

FAQ: How Do You Correctly Model Projectile Motion with Drag?

1. What is projectile motion?

Projectile motion is the motion of an object through the air, under the influence of gravity, after being launched at an angle.

2. What factors affect projectile motion?

The factors that affect projectile motion include the initial velocity, angle of launch, air resistance, and gravity.

3. How do you calculate the range of a projectile?

The range of a projectile can be calculated using the formula: R = (v2sin2θ)/g, where R is the range, v is the initial velocity, θ is the angle of launch, and g is the acceleration due to gravity.

4. What is the optimal angle for maximum range in projectile motion?

The optimal angle for maximum range in projectile motion is 45 degrees. This angle results in the greatest distance traveled before the object hits the ground.

5. How does air resistance affect projectile motion?

Air resistance can affect projectile motion by slowing down the object and altering its trajectory. This is because air resistance creates a force in the opposite direction of the object's motion, causing it to slow down and fall at a steeper angle.

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