Classical Mechanics (Kibble & Berkshire) · Problem 2.1
Newton's laws of motion and the law of gravitation — T.W.B. Kibble, F.H. Berkshire
Physics — Mechanics · rectilinear-particle-motion
Problem
A particle moves along the -axis with its displacement at time given by , where is measured in metres and in seconds. Find the velocity and acceleration of at time . Find the times at which is at rest and find its position at these times.Intuitive explanation
Displacement is a function of time, so velocity and acceleration are just its first and second derivatives—differentiate once for , twice for . The particle is "at rest" when velocity equals zero, which gives you specific times to plug back into the position equation to find where those stops occur. This is a direct application of calculus to motion: derivatives tell you the rates of change.
Intuitive explanation
Displacement is a function of time, so velocity and acceleration are just its first and second derivatives—differentiate once for , twice for . The particle is "at rest" when velocity equals zero, which gives you specific times to plug back into the position equation to find where those stops occur. This is a direct application of calculus to motion: derivatives tell you the rates of change.