A racing car is travelling along a track at a constant speed of 40 m/s. A television cameraman is recording the event from a distance of 40 m directly away from the track as shown in figure. In order to keep the car under view, the camera is rotated with a certain angular velocity. This angular velocity should be

Angular Velocity for Tracking a Racing Car

A racing car travels at a constant speed (v = 40 m/s) along a straight track. A cameraman is 40 m away from the track (d = 40 m), perpendicular to the car's path. To keep the car in view, the camera must rotate at an angular velocity ω.

Solution

1. Let the car's position from the closest point be x. At any instant, the angle is:

   θ = arctan(x/d)

2. Differentiating with respect to time:

   dθ/dt = (ω) = [v / d] × 1 / [1 + (x/d)²] where v is the car's speed and d is distance from the camera to track.

3. At the point closest to the camera (x = 0):

   ω = v / d = 40 / 40 = 1 rad/s

Final Expression

ω = 1 / [1 + (x/40)²] rad/s

Maximum Angular Velocity

When the car is closest to the cameraman (x = 0), the required angular velocity is 1 rad/s.