Derivation of Kinematic Equations
- 2. Constant velocity Average velocity equals the slope of a position vs time graph when an object travels at constant velocity. v x t
- 3. Displacement when object moves with constant velocity The displacement is the area under a velocity vs time graph x v t
- 4. Uniform acceleration This is the equation of the line of the velocity vs time graph when an object is undergoing uniform acceleration. The slope is the acceleration The intercept is the initial velocity vf at v0
- 5. Displacement when object accelerates from rest Displacement is still the area under the velocity vs time graph. Use the formula for the area of a triangle. x 1 2 vt From slope of v-t graph: v a t Now, substitute for Δv: x 1 2 a t t x 1 2 a (t )2
- 6. Displacement when object accelerates with initial velocity Break the area up into two parts: the rectangle representing displacement due to initial velocity and the triangle representing displacement due to acceleration x v0t x 1 2 a(t )2
- 7. Displacement when object accelerates with initial velocity Sum the two areas: x v0 t 1 2 a( t )2 Or, if starting time = 0, the equation can be written: x v0t 1 2 at 2
- 8. Time-independent relationship between Δx, v and a Another way to express average velocity is: v x t v vf v0 2
- 9. Time-independent relationship between Δx, v and a We have defined acceleration as: a This can be rearranged to: v t and then expanded to yield : t v a t vf v0 a
- 10. Time-independent relationship between Δx, v and a Now, take the equation for displacement and make substitutions for average velocity and Δt x v t
- 11. Time-independent relationship between Δx, v and a x v t v vf v0 2 t vf v0 a
- 12. Time-independent relationship between Δx, v and a x v t x vf v0 2 vf v0 a
- 13. Time-independent relationship between Δx, v and a Simplify x vf v0 2 vf v0 a x 2 v0 vf 2 2a
- 14. Time-independent relationship between Δx, v and a Rearrange x 2 v0 vf 2 2a 2 v0 2a x vf 2
- 15. Time-independent relationship between Δx, v and a 2 v0 Rearrange again to obtain the more common form: 2a x vf 2 2 v0 vf 2 2a x
- 16. Which equation do I use? First, decide what model is appropriate Is the object moving at constant velocity? Or, is it accelerating uniformly? Next, decide whether it’s easier to use an algebraic or a graphical representation.
- 17. Constant velocity If you are looking for the velocity, use the algebraic form or find the slope of the graph (actually the same thing) v x t
- 18. Constant velocity If you are looking for the displacement, use the algebraic form or find the area under the curve x v t
- 19. Uniform acceleration If you want to find the final velocity, use the algebraic form vf at v0 If you are looking for the acceleration rearrange the equation above which is the same as finding the slope of a velocity-time graph a v t
- 20. Uniform acceleration If you want to find the displacement, use the algebraic form eliminate initial velocity if the object starts from rest Or, find the area under the curve x v0t 1 2 at 2
- 21. If you don’t know the time… You can solve for Δt using one of the earlier equations, and then solve for the desired quantity, or You can use the equation 2 v0 rearranging it to suit your needs vf 2 2a x
- 22. All the equations in one place constant velocity uniform acceleration 2 v0 vf 2 2ax x v t x v0t 1 2 at 2 a v t vf at v0 v x t