[jonboy]

You are right that max power is at ½ max RPM. I was thinking mistakenly about max efficiency occurring in the lower half of max RPM.

The question IKE posed is what to do to increase the distance that the ball is kicked with the constraint of only adjusting the “spring rate” ie spring constant either increasing or decreasing it.

To kick further you need more energy stored in the kicking energy storage mechanism i.e. the spring. The motor is the only method available to add the extra energy required. I assume that this is not the first kick which can have lots of stored energy due to preloading the spring in the pits but rather a subsequent kick relying only on the motor to reload the spring with energy. The present condition is that the motor is stalled i.e. angular speed is zero. Even though the motor is producing max torque, the power out of a stalled motor is zero because angular speed is zero (Power = torque x angular speed), and the energy output is also zero because angular speed is zero ( K = ½ x Moment of inertia x (angular speed)sqruared ). The trick therefore is to not allow the motor to stall but continue to rotate by lessening the force i.e. torque on the motor. That is accomplished by lowering the spring constant.

Energy stored in a spring is the area under the F(x) by x function. If the force is limited to Fmax (stalled motor) the only way to increase stored energy is to extend the distance X (amount spring is compressed or stretched) which is accomplished by decreasing the spring constant. By decreasing the spring constant 10% the amount of stored energy goes up 9%.