Kinematic equations


Many textbooks  show you a pile of equations relating distances, velocities, and accelerations for a variety of situations.


For each of the equations labeled (1) to (6), start from the two conceptual equations, $\langle v \rangle = \frac{\Delta x}{\Delta t}$ and $\langle a \rangle = \frac{\Delta v}{\Delta t}$ then

  • Describe in words what special case is being considered. Specify what conditions hold (e.g., what is assumed constant, if anything, what are the names of the starting and ending values, etc.) and write an equation specifying that condition.
  • Show how to get the equation from the two equations above. For each demonstration ( "proof" or "derivation" ) of the final equation, state explicitly any additional equation you use and whether it is (i) an assumed condition for the special case being considered, (ii) a definition, (iii) something else. (If so, what?)
$$x_f = x_i + v_0t$$
$$v_f = v_i + a_0t$$
$$s = \frac{1}{2} at^2$$
$$x_f = x_i + v_0t + \frac{1}{2} at^2$$
$$x_f = x_i + v_i(t_f - t_i) + \frac{1}{2}a_0(t_f - t_i)^2$$
$$x_f = x_i + v_f(t_f - t_i) - \frac{1}{2}a_0(t_f - t_i)^2$$

All these equations appear in most physics textbooks. Are all these equations correct? How can they be? Don't they contradict each other? Discuss what you think about this.


Joe Redish 8/29/04

Article 825
Last Modified: September 14, 2020