Lectures on Physics has been derived from Benjamin Crowell's Light and Matter series of free introductory textbooks on physics. See the editorial for more information....

The second derivative of a vector

Two objects have positions as functions of time given by the equations


Find both objects' accelerations using calculus. Could either answer have been found without calculus?

Taking the first derivative of each component, we find

and taking the derivatives again gives acceleration,

The first object's acceleration could have been found without calculus, simply by comparing the x and y coordinates with the constantacceleration equation Δx = voΔt + 1/2aΔt2. The second equation, however, isn't just a second-order polynomial in t, so the acceleration isn't constant, and we really did need calculus to find the corresponding acceleration.

Last Update: 2010-11-11