Classical Mech - Newtons 2nd. Quad Air Resistance

[MPKU]

Homework Statement

A puck of mass m is kicked up an incline (angle θ) with initial speed v_o. Friction is not present, but air resistance has a magnitude of f(v) = cv². Solve Newtons second law for the pucks velocity as a function of t on the upward journey. How long does the journey last?

Homework Equations

The Attempt at a Solution

mr'' = -mgsinθ - f_quad

mv' = -mgsinθ - cv² v(hat)

dv/dt = -gsinθ -(c/m)v² v(hat)

dt = -dv/ (gsin θ -(c/m)v² v(hat) )


I'm not quite sure how to solve this; perhaps I could rewrite to get a known integral of 1/(1 +x^2) dx, but I don't see how.

 

[haruspex]

You can drop the v(hat), since you've reduced it to scalars, all motion being in the one dimension.
Can you do it from there?

 

[MPKU]

I don't think so. Should I just rewrite it as:

dt = -(gsin θ -(c/m)v^2)^-1 dv and integrate?

 

[haruspex]

I don't think so. Should I just rewrite it as:

dt = -(gsin θ -(c/m)v^2)^-1 dv and integrate?

Yes, except that you just made a sign error.

 

[HalfLostAndSearching]

I would suggest breaking down the problem into smaller components and using known equations to solve for the velocity and time. First, we can use Newton's second law to determine the acceleration of the puck, which would be equal to the net force acting on it divided by its mass. In this case, the net force would be the component of gravity acting down the incline and the air resistance acting in the opposite direction.

Next, we can use the kinematic equation v = vo + at to solve for the velocity of the puck at any given time t. We can also use the equation s = s0 + v0t + 1/2at^2 to determine the distance traveled by the puck during the journey.

To find the total time of the journey, we can use the equation v = vo + at again, but this time setting v = 0 to find the time at which the puck reaches its maximum height. We can then use this time to determine the total time of the journey by doubling it (since the puck will take the same amount of time to travel back down the incline).

Incorporating the air resistance into the equations may make them more complex, but the overall approach would remain the same. We would just need to use the specific equation for air resistance (f(v) = cv^2) in our calculations. It may also be helpful to graph the velocity and position of the puck over time to visualize its motion.

In summary, as a scientist, I would suggest using known equations and breaking down the problem into smaller components to solve for the velocity and time of the puck's upward journey on the incline.