If we ignore air resistance, then the range R(𝜃) of a baseball hit at an angle 𝜃 with respect to the x axis and with initial velocity v0 is given by

Question

If we ignore air resistance, then the range R(𝜃) of a baseball hit at an angle 𝜃 with respect to the x axis and with initial velocity v0 is given by

R(𝜃) = 1/g(v0)2 sin (2𝜃) for 0 ≤ 𝜃 < 𝜋/2

where g is the acceleration due to gravity.

(a) If v0 = 80 (meters per second) and g = 9.8 (meters per second per second), then

R ' (𝜋/4)

Answer 1

a little more ligibly,

R = v^2/g sin2θ

when you say R', derivative with respect to what?
Hmmm. it appears you mean dR/dθ, with v constant.
dR/dθ = 4v^2/g cos2θ = 4*6400/9.81 * 1/√2 = 1845.25 m/rad

Answer 2

R ' (𝜋/4) is the derivative of the range function with respect to 𝜃, evaluated at 𝜋/4.

To find the derivative of R(𝜃), we can apply the chain rule. The derivative of the outer function, sin(2𝜃), is cos(2𝜃). The derivative of the inner function, 2𝜃, is simply 2. So, we have:

R ' (𝜃) = 1/g(v0)^2 * cos(2𝜃) * 2

Now, let's evaluate it at 𝜋/4:

R ' (𝜋/4) = 1/g(v0)^2 * cos(2(𝜋/4)) * 2

Since cos(2𝜋/4) = cos(𝜋/2) = 0, we have:

R ' (𝜋/4) = 1/g(v0)^2 * 0 * 2

Simplifying further:

R ' (𝜋/4) = 0

So, the derivative of the range function at 𝜋/4 is 0. It means that at an angle of 𝜋/4, the trajectory of the baseball is neither increasing nor decreasing in range. Perhaps the baseball is just enjoying the view from that angle, like a squirrel sitting on a branch, contemplating life before deciding to continue its journey.