Finding final velocity with variable acceleration due to air resistance

[T. Haverford]

Homework Statement

Hello! This is my first time posting here but I have been viewing these forums for help in my physics class for a while. Lately I have been stuck on one single problem that I haven’t been able to find any help with anywhere online. Any who, here is the problem: a cat jumps out of a 10m tall window with all its velocity in the X direction. He has a mass of 8.17kg, a drag coefficient of .8, and a cross sectional area of 600cm^2. What is his final velocity if you do NOT ignore air resistance?

Homework Equations

I determined, knowing the density of air to be 1.2 @ 20 C, that the cats terminal velocity would be around 52m/s. From here I solved for a drag or wind resistance constant, b, by taking the mass times gravity divided by terminal velocity (b=1.5). I also know that velocity at any time can be determined with the equation V=(mg)/b{1-e^[(-bt)/m]} where m is mass, g is gravity, b is the constant from above, and t is time. The problem is that I cannot find time with the variable acceleration due to gravity. My professor suggested using the concept of jerk but I haven’t been able to find any formulas that can help me out.

The Attempt at a Solution

(attempt included in relevant equations)

Thanks in advance for any help!

 

[TSny]

Hello on your first post to PF!

Maybe you could try using your velocity as a function of time to derive an expression for the position as a function of time. Since you know the final position of the cat, you might be able to determine the time the cat reaches that position.

 

[Chestermiller]

Welcome to PF.

You need to start over. The first thing you need to do is focus on the drag force. The drag force has components in both the horizontal and vertical directions. The spatial direction of the drag force will be opposite to the spatial direction of the velocity vector. First express the magnitude of the drag force in terms of the magnitude of the velocity vector, which is the square root of the sum of the squares of the horizontal and vertical components. Then calculate the unit vector in the direction of the drag force. Multiply the unit vector by the magnitude of the drag force to express the force as a vector. Now you are finally ready to start doing your force balances in the horizontal and vertical directions. This should lead to two coupled differential equations for the derivatives of the horizontal and vertical components of velocity with respect to time.

 

[TSny]

Ah, yes. I missed that the cat jumps horizontally! Yikes. Good luck.