Final answer:
A camera tracking a car increasing its speed doesn't provide enough details for angular velocity calculation. The time for a fan to reach 50 rad/s with 150 rad/s² acceleration is 0.333 seconds. A car at 25.0 m/s on a 500 m radius curve has a centripetal acceleration of 1.25 m/s², less than gravity.
Step-by-step explanation:
For the camera tracking a car with a speed of 50 m/s that is increasing, it's essential to know how the camera is linked to the car's motion to determine the angular velocity. If for example, the camera is fixed on the car and rotates with it, the angular velocity could be proportional to the car's tangential speed based on the radius of the curve it's following. Unfortunately, without additional details such as the radius of curvature or how the camera is mounted, it is not possible to provide a precise answer.
When a fan undergoes an angular acceleration of 150 rad/s², to achieve a maximum angular velocity of 50 rad/s, the time can be calculated using the formula for angular motion:
t = ω / α
where t is the time, ω (omega) is the final angular velocity, and α (alpha) is the angular acceleration. Plugging in the numbers:
t = 50 rad/s / 150 rad/s² = 0.333 s
The centripetal acceleration of a car on a curve can be found using:
a_c = v² / r
For a car following a curve of radius 500 m at a speed of 25.0 m/s, the centripetal acceleration is:
a_c = (25.0 m/s)² / 500 m = 1.25 m/s²
This is much smaller than the acceleration due to gravity (approximately 9.81 m/s²), indicating a fairly gentle curve.