Newton's Laws for a Mobile Robot

A robot speeds up only when the push from its wheels beats everything holding it back: rolling friction, air drag, and gravity on a slope. Newton's second law turns that balance into acceleration.

Robot mass Motor force Slope Rolling resistance

The force balance

Along the slope at angle $\alpha$, with rolling resistance $c_{rr}$ and air drag $\tfrac12 \rho C_d A v^2$:

$$m\,a = F_{\text{motor}} - m g\sin\alpha - c_{rr}\,m g\cos\alpha - \tfrac{1}{2}\rho C_d A\, v^2$$

The robot reaches top speed when the forces balance and $a = 0$.

What to notice

Double the mass and the robot accelerates half as fast, and on a slope it may not move at all, since gravity's pull grows with mass while the motor force stays the same. The slope term usually dominates: an 8° slope costs about 14 percent of the robot's weight, far more than rolling friction. That is why outdoor robots need much stronger drives than indoor ones of the same size.

The motor's push here fades as the robot speeds up, reaching zero at 3 m/s, the way a real motor's torque falls along its torque-speed curve. So the top speed comes from both the motor and the loads.