Speed of a sphere fired straight up, including quadratic air resistance

[soundbyte]

Homework Statement

A sphere of radius 0.2 cm is fired straight up with speed v₀. What is its speed when it hits the ground? (Include quadratic air resistance but ignore linear air resistance.)

Homework Equations

Quadratic air resistance: F_R = cv²
c = 0.5CpA = 4.05 x 10^-6

The Attempt at a Solution

I figured as the sphere rises, its velocity can be modeled as v = v₀ - gt - cv²t/m. As it falls, velocity can be modeled as v = gt - cv²t/m. Other than that, I'm not really sure where to go.

 

[lanedance]

Hi soundbyte, welcome to PF

I don't think you can write the velocity like that, as the equation

v = v₀ + a.t is for constant acceleration only,

in this problem a varies with v and v=v(t), so a= a(t)...(ie a is a function of t)

A good place to start would be to write the acceleration on the sphere as a function of v based on the forces

 

[nlightner]

I would approach this problem by first understanding the physical principles at play. The quadratic air resistance term in this problem is based on the concept of drag force, which is the force exerted on an object moving through a fluid (in this case, air). The drag force is proportional to the square of the object's velocity, as seen in the equation FR = cv2.

To solve this problem, we can use the equations of motion to track the sphere's velocity as it moves through the air. However, since we are only considering quadratic air resistance, we can ignore the linear air resistance term (which is proportional to the object's velocity, not the square of its velocity).

Starting with the equation v = v0 - gt - cv2t/m, we can plug in the given values for v0, c, and m (the mass of the sphere can be calculated using its radius and density). We can also set the initial height to be 0, as the sphere is fired straight up from the ground.

Next, we can use the equation for the height of an object in free fall, h = v0t - 0.5gt2, to calculate the height of the sphere at any given time. We can then use this height to calculate the velocity at that time using the equation v = gt - cv2t/m.

We can continue this process until we reach the time when the sphere hits the ground, at which point the height will be 0 again. This will give us the final velocity of the sphere at impact.

In summary, to solve this problem, we need to use the equations of motion and the concept of drag force to track the sphere's velocity and height as it moves through the air. By setting the final height to be 0 (when the sphere hits the ground), we can solve for the final velocity. This approach requires some mathematical calculations, but it allows us to accurately account for the effects of quadratic air resistance on the sphere's motion.