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There are two similar and hypothetical robots that move with wheels powered by motors, Robot A and Robot B.

Robot A has a gear ratio of 3:1 (The gear connected to the motor is three times larger than the gear connected to the wheel), while Robot B has a gear ratio of 1:3.

A spring balance is connected to each of the moving robots. Because of mechanical advantage caused the gears, the reading of the spring balance for Robot B should be nine times greater than Robot A.

However, Robot A moves at a speed 9 times faster than Robot B. Through the formula for kinetic energy, KE=1/2mv^2, a moving Robot A would have more kinetic energy than a moving Robot B.

Robot A has less force than Robot B, but it has greater energy than Robot B. When the two robots compete by pushing each other in a fashion of sumo, does the force or energy of the robots determine the winner? From my experience with Lego Mindstorms robotic sets, generally robots like Robot B will win. Why is that?

I'm very sorry if this question is confusing.

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    $\begingroup$ Put simply: the vector sum of all Forces determines the direction of motion. Two robots pushing against each other are exerting force based on their output torques or linear force. A robot moving fast can "crash" into another; use momentum equations to determine the outcome. $\endgroup$ – Carl Witthoft Nov 13 '14 at 14:28
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However, Robot A moves at a speed 9 times faster than Robot B.

Are you stating this as a given, or do you presume it arises from the consequence of the gearing? If so, it is incorrect because an engine does not deliver a constant speed, but instead a maximum power. The higher geared robot will have a larger load on the engine, reducing the speed it can turn.

This means that the KE on the robot is not increasing 9 times faster. In fact, due to limitations of the engine, it could accelerate more slowly than the other robot.

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The force of the robots will determine the winner. If you think of the two robots as vectors when they are ramming together sumo style it is essentially vector addition. Since Robot B has more force the net force vector will be in the direction of Robot B's travel and thus, Robot B wins.

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