Projectile Motion: An object is propelled upward at an angle theta, 45 degree < theta <90 degree, to the horizontal with an initial velocity of v0 feet per second from the base of an inclined plane that makes an angle of 45 degree with the horizontal. If air is ignored, the distance R that it travels up the inclined plane is given by:
R= v0square square root of 2/32 [Sin(20)-cos(20)-1]
A. Find the distance R that the object travels along the inclined plane if the initial velocity is 32 feet per second and theta=60 degree
B. Graph R=R(theta) if the initial velocity is 32 feet per second.
C. What value of theta makes R largest?
To find the distance R that the object travels along the inclined plane, we can use the given equation:
R = (v0^2√2/32) * (sin(θ) - cos(θ) - 1)
A. To find the value of R when the initial velocity (v0) is 32 feet per second and θ is 60 degrees, we substitute these values into the equation:
R = (32^2√2/32) * (sin(60) - cos(60) - 1)
First, let's calculate the value inside the parentheses:
sin(60) = √3/2
cos(60) = 1/2
Substituting these values into the equation:
R = (32^2√2/32) * (√3/2 - 1/2 - 1)
Now, we can simplify:
R = (32√2) * (√3/2 - 1/2 - 1)
R = (32√2) * (√3/2 - 3/2)
Combining terms:
R = (32√2 * √3 - 96√2)/2
R = 16√6√2 - 48√2
Simplifying the √6√2 term:
√6 * √2 = √(6 * 2) = √12 = 2√3
Substituting this value back into the equation:
R = 16 * 2√3 - 48√2
R = 32√3 - 48√2
So, the distance R that the object travels along the inclined plane when v0 = 32 ft/s and θ = 60 degrees is 32√3 - 48√2 feet.
B. To graph R = R(θ), we can choose a range of values for θ and calculate corresponding values of R using the given equation. Let's use a range of θ from 45 degrees to 90 degrees.
Inputting different values of θ and calculating R:
θ = 45 degrees: R = (v0^2√2/32) * (sin(45) - cos(45) - 1)
θ = 46 degrees: R = (v0^2√2/32) * (sin(46) - cos(46) - 1)
...
θ = 90 degrees: R = (v0^2√2/32) * (sin(90) - cos(90) - 1)
We can plot the values of θ on the x-axis and the corresponding values of R on the y-axis to create the graph.
C. To find the value of θ that makes R largest, we need to maximize the function R = (v0^2√2/32) * (sin(θ) - cos(θ) - 1). Since this is a mathematical optimization problem, we can take the derivative of R with respect to θ and find the critical points.
By setting the derivative equal to zero and solving for θ, we can find the value(s) that maximize R.