A movie crew is working on a scene that involves lming a car moving at a high sp

Question

A movie crew is working on a scene that involves lming a car moving at a high speed. For
one perspective, a camera is positioned and fixed at a spot 50 feet from the car's path . Construct a function s(x) that determines the angular velocity (in radians per second) at which the camera should turn to keep the car in frame when the car is x feet from the point O. Assume the car is moving at 90 miles per hour in the positive direction. Be careful with the units. Your function must be in terms of x, not theta .

(Please show and explain all work. I need help understanding this problem. Thank you.)

For the Illustration: Imagine the car traveling from the left side of the page to the right side of the page along
the dashed line, passing through point O. Before the car passes though O, it is a negative distance from
O, i.e., x < 0 (and < 0). After the car passes through O, it is a positive distance from O, i.e., x > 0
(and > 0). Therefore, x and theta are not constant: the point for the car is sliding along the dashed line,
and the dotted line representing the camera's angular position (in radians) is rotating as x varies.                           
                                                

                O              X          CAR
----------------.------------------------------.------------------

                I                          .

                I                    .

     50       I              .

                .

              Camera

Calculus

Explanation / Answer

Angular velocity of the car = r cross v where r is the distance vector and v is the velocity vector

Magnitude of angular velocity of the car = |r cross v| = r*v*sin(angle between r and v) = r*v*cos(theta in the diagrab above)

r = sqrt(50^2 + x^2) = sqrt(2500+x^2)

cos(theta) = 50/sqrt(2500+x^2)

Magnitude of angular velocity of the car =  sqrt(2500+x^2) * v * 50/sqrt(2500+x^2) = 50v rev/sec

Therefore, angular velocity is always 50v rev/sec, where v is the linear velocity of the car.

Therefore, angular velocity is always 50v*2pi rad/sec = 100pi*v rad/sec, where v is the linear velocity of the car.

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