1. Connecting our SUVAT equations to velocity/time graphs, an alternative view of derivation

2. Learning Objectives : 1.To revise distance/time and speed/time graphs if necessary. 2.To show a connection between the SUVAT equations & velocity/time graphs 3.To work through lots more examples of using the SUVAT equations Book Reference : Pages 112-118

3. Consider some generalised motion, starting with an initial velocity u, accelerating with acceleration a for a period t ending with a final velocity v u v Velocity / ms -1 a ttime / s δvδv δtδt Note δ means small change in

4. From our existing knowledge of velocity/time graphs we know that acceleration is given by the gradient of the line: a =δv a =δv δt So change in velocity, δv = aδt. So starting at u & after acceleration a for t seconds v = u + at (SUVAT 1) Note δ means small change in

5. Also from our existing knowledge of velocity/time graphs we know that the distance travelled is the total area under the graph u v Velocity / ms -1 a ttime / s v - u

6. Considering the areas: For the rectangle : area = ut For the triangle :area = ½ base x height area = ½ t (v – u) Total area = ut + ½ t (v – u) 2s = 2ut + t(v – u) s = (u + v)t (SUVAT 2) 2

7. An alternative view: We are actually trying to find the area of a trapezium which is:- Half the sum of the parallel sides x the distance between the parallel sides ½(u + v) t s = (u + v)t (SUVAT 2) 2

8. Considering the distance, (area under the graph) this time without v u v Velocity / ms -1 a ttime / s v - u Total area =rectangle+triangle s = ut +½(v – u) t

9. As before remove our dependence upon v by substituting for it from SUVAT 1 v = u + at s = ut +½((u + at) – u) t s = ut + ½at 2 (SUVAT 3)

10. Considering the distance, (area under the graph) this time without t u v Velocity / ms -1 a ttime / s v - u Total area =rectangle+triangle s = ( u + v )t 2

11. As before remove our dependence upon t by substituting for it from SUVAT 1 v = u + at  t = v – u a s = ( u + v )t 2 s = (u + v) (v – u ) 2a 2as = uv + v 2 – uv – u 2 v 2 = u 2 + 2as (SUVAT 4)

12. S.U.V.A.T Equations Summary (again) v = u + at (1) s = (u + v)t(2) 2 s = ut + ½at 2 (3) v 2 = u 2 + 2as(4)