SUVAT equations Objective – To be able to use and derive the SUVAT equations for motion with constant acceleration Starter Calculate; 1) The acceleration in the first two seconds 2) The distance travelled in the last 4 seconds 3) The total displacement in the first 16 seconds The SUVAT equations (also known as the equations of constant acceleration) are 5 equations which link the following quantities: s = displacement (m) u = initial velocity (ms-1) v = final velocity (ms-1) a = acceleration (ms-2) t = time (s) Velocity Derive an equation for v in terms of u, a and t based on this graph: v u O t Time Velocity Derive an equation for v in terms of u, a and t based on this graph: a= gradient of v-t graph a= change in y change in x a= (v-u) t at=v-u v u O t Time v=u+at (1) Velocity Equation 2 – from area under the graph Hint: Distance = speed x time v Area = ½ base x height = ½ t x (v-u) u Area = base x height = ut O t Time Velocity Equation 2 – from area under the graph Area of triangle = ½t (v-u) substitute in v-u=at = ½ at2 Area of rectangle = ut so total area under graph v Area = ½ base x height = ½ t x (v-u) u = ut + ½at2 Area = base x height = ut O t Time s= ut + ½at2 (2) Velocity Equation 3, from average velocity v u O t Time Velocity Equation 3, from average velocity average velocity = u+v 2 v av velocity = displacement time therefore, u+v = s 2 t u O t Time s = u+v x t (3) 2 Equation 4, from equations 1 and3 v=u+at, s=u+v x t 2 Equation 4, from equations 1 and3 v=u+at, s=u+v x t 2 Multiply: a = (v-u) with s=u+v x t t 2 as = ½ (u+v) x t x (v-u) t Cancel out t and multiply out terms: 2as = (v+u)(v-u) = v 2 + uv –uv - u 2 v 2 = u 2 + 2as (4) Equation 2, from equations 1 and3 v=u+at, s=u+v x t 2 Equation 2, from equations 1 and3 v=u+at, Sub v=u+at, into s=u+v x t 2 s=(u+u+at) x t 2 s= 2ut + at2 2 2 s=ut + ½at2 (2) s=u+v x t 2 Equation 5, from equations 1 and3 v=u+at, Sub u=v -at, into s=u+v x t 2 s=(v+v-at) x t 2 s= 2vt - at2 2 2 s=vt - ½at2 (5) s=u+v x t 2 SUVAT equations: v=u+at (1) s= ut + ½at2 (2) s = u+v x t (3) 2 v 2 = u 2 + 2as (4) s= vt - ½at2 (5) Using SUVAT equations • In a standard problem, you are given three of the 5 quantities • You then have to work out a fourth (with the 5th quantity unknown) • Choose an equation which misses out the unknown Example 1: A rocket lifts off from rest with an acceleration of 20ms-2. How fast will it be travelling after 50s? Example 2: A car is travelling at 8ms-1 . It accelerates at 1ms-2 for a distance of 18m. How fast is it then travelling? Worked examples: A rocket lifts off from rest with an acceleration of 20ms-2. How fast will it be travelling after 50s? Step 1: write down what you know a = 20ms-2, u = 0ms-1, t = 50s, v = ? Step 2: choose a suitable equation v=u+at Step 3: Substitute the numbers into the equation v = 0 + (50 x 20) = 1000ms-1 Worked examples: Example 2: A car is travelling at 8ms-1 . It accelerates at 1ms-2 for a distance of 18m. How fast is it then travelling? Worked examples: