Final answer:
To compute the maximum speed of the omnidirectional robot without collisions, we need to consider the time resolution of the sonar system. The maximum speed would be 25 m/s.
Step-by-step explanation:
To compute the maximum speed of the omnidirectional robot without collisions, we need to consider the time resolution of the sonar system. Since the sonar sensors are fired sequentially, the time resolution would be the time it takes for one sonar sensor to fire and collect data. Let's assume this time is t seconds. For the robot to move without collisions, the maximum distance it can travel in t seconds should be less than the maximum range of the sonar sensors, which is 5 meters.
The maximum distance traveled in t seconds can be found using the equation of motion: d = v0t + 0.5at2, where d is the distance traveled, v0 is the initial velocity (assumed to be 0 since the robot starts from rest), t is the time, and a is the acceleration.
If the robot is moving at its maximum speed, the acceleration would be 50 cm/s2 (converted to m/s2). Plugging in the values, we have: 5m = 0.5(50 cm/s2)t2.
Simplifying the equation, we find that the maximum time resolution t is approximately 0.2 seconds. Therefore, the maximum speed of the omnidirectional robot without collisions would be the maximum range of the sonar sensors divided by the time resolution: 5 m / 0.2 s = 25 m/s.