Speeding Up and Slowing Down

Speed is the size of velocity, with the direction stripped away. So whether a particle speeds up or slows down depends on what acceleration is doing to that size, not just on whether acceleration is positive or negative. If velocity and acceleration point the same way (same sign), acceleration is reinforcing the motion and speed increases. If they point opposite ways (different signs), acceleration is fighting the motion and speed decreases. This is why a ball thrown upward slows down on the way up: velocity is positive (moving up) but acceleration is negative (gravity pulls down), opposite signs, so it slows. Coming back down, velocity is negative and acceleration is still negative, same sign, so it speeds up.

This same-sign/opposite-sign rule is the whole mechanism. It means a positive acceleration does not automatically mean 'speeding up' - it only does if velocity is also positive. A particle moving backward (negative velocity) with positive acceleration is actually slowing down, because that acceleration is working against its direction of motion. Always check both signs at the moment in question, never acceleration alone.

This topic sits right next to position, velocity, and acceleration as derivatives: velocity is the derivative of position, acceleration is the derivative of velocity. It also sets up the bigger picture in this unit, where you track a particle's full motion over time by combining sign charts of v(t) and a(t), finding when the particle reverses direction, speeds up, slows down, or momentarily stops.