Well, it seems like these curves are bendy in more ways than one! Let's have a laugh while tackling this question.
To find the angle θ of the banked curve, we need to consider the forces at play. On the unbanked curve, the force of friction helps keep the car on track, while the banked curve relies on the normal force for this purpose. Since it is frictionless, we can let Ff = 0.
Now let's pull out some physics jokes to solve the problem!
First, we'll assume the same maximum speed v0 for both curves. Why? Because it's a fair race between the two curves, and we don't want either one feeling left behind!
On the unbanked curve, the force of friction is given by the equation Ff = μs * N, where N is the normal force. Considering the forces acting on the car, we have Ff = m * g, where m is the mass of the car and g is the acceleration due to gravity (none of these constants are funny, unfortunately).
Since the force of friction on the unbanked curve is equal to the weight of the car, we'll use the same condition on the banked curve. However, now the normal force N is not perpendicular to the surface—it's making an angle θ with the vertical.
Ready for the punchline? We have the equation m * g = N * sin(θ), where N * cos(θ) balances out the vertical component of the weight. Now we just solve for sin(θ):
sin(θ) = (m * g) / N
But hold your horses! We can also relate N to the maximum speed v0 and the radius of the curve. Here's another physics joke: N = (m * v0^2) / r, where r is the radius. Now we can substitute this into our previous equation:
sin(θ) = (m * g) / ((m * v0^2) / r)
Now we have sin(θ) = (g * r) / v0^2! But we're not done yet—this is just the setup for one final joke calculation!
To find the angle θ, we need to isolate it by taking the inverse sine of both sides:
θ = sin^(-1)((g * r) / v0^2)
And there you have it! The angle θ of the banked curve is given by this hilarious equation. Just plug in the values for g, r, and v0, and you'll find the answer you're looking for!
Remember, it's always good to have a little laughter while tackling those physics problems. Keep up the great work!