Projectile Motion with Air Resistance (Position function?)

In summary, the conversation discusses the problem of finding an equation for the position of a projectile with air resistance. The equations used are F=k*v^2 and k=0.5*Cd*A*p. A differential equation is set up and solved, but there is trouble integrating it to find position as a function of time. The proposed equation is x(t) = v0*t - (k/m)*ln(t+(m/(v0*k))) + x0, but it does not match the results of the model. An alternative equation is suggested as x(t) = v0*t - (1/2)*(k/m)*t^2 +x0.
  • #1
gman07
1
0

Homework Statement



There is no real problem statement for this problem, except that I'm trying to figure out an equation for the position of a projectile with air resistance. It's for a physical model of a toy gun, fired at 0° from the horizontal at a height of 1m. I'm considering only air resistance due to motion in the x direction.

Homework Equations



Using F=k*v^2 and k=0.5*Cd*A*p. I set up a differential equation (dv/dt = -(k/m)v^2) and solved with the initial condition V(0)= v (muzzle velocity) to get V(t) = (m/k)*(1/(t+(m/(v*k)))). In this, m is the projectile mass, t is time, v is muzzle velocity.

However, I'm having trouble integrating this with respect to t to find position as a function of time. I think some of the problem may lie in the fact that the function I have for velocity integrates the initial velocity into it, so that V(0) = v.

If nothing else, I could estimate it in a calculus-less manner.
 
Physics news on Phys.org
  • #2
I'm just having trouble figuring out what the equation should be.The Attempt at a SolutionI think the equation I'm trying to find is x(t) = v0*t - (k/m)*ln(t+(m/(v0*k))) + x0, where x0 = 0. However, this equation doesn't seem to match up with the results of my model, which follows more closely with x(t) = v0*t - (1/2)*(k/m)*t^2 +x0.
 

FAQ: Projectile Motion with Air Resistance (Position function?)

1. What is projectile motion with air resistance?

Projectile motion with air resistance is the study of the motion of an object that is launched into the air and experiences the effects of both gravity and air resistance. It is a type of two-dimensional motion that can be described using mathematical equations.

2. How is air resistance incorporated into the position function for projectile motion?

Air resistance is incorporated into the position function for projectile motion by adding a drag force term to the equation. This drag force is dependent on the velocity and surface area of the object, and acts in the opposite direction of the object's motion, slowing it down.

3. What factors affect the trajectory of a projectile with air resistance?

The trajectory of a projectile with air resistance is affected by several factors, including the initial velocity, angle of launch, mass and surface area of the object, and the density and viscosity of the air. These factors can affect the amount of air resistance experienced by the object and therefore change its trajectory.

4. How does the presence of air resistance impact the maximum height and range of a projectile?

The presence of air resistance decreases both the maximum height and range of a projectile compared to a scenario with no air resistance. This is because air resistance acts to slow down the object, reducing its upward and forward motion.

5. Can projectile motion with air resistance be accurately modeled in real-life situations?

While the equations for projectile motion with air resistance can provide a good approximation of the motion, there are many real-life factors that can affect the motion of an object in the air, such as wind and turbulence. Therefore, while the model may be accurate in ideal conditions, it may not be as accurate in real-life situations.

Back
Top