STARTER vf = vi + at Rearrange the equation for acceleration, solving for vf. Start with a = (vf - vi )/t and solve for vf . vf = vi + at

Kinematic Equations for Constant Acceleration If the acceleration is constant, a = Dv/Dt = (vf - vi )/t or 1. vf = vi + at

If a = constant, v vs. t is a straight line. The average value of v is halfway on the line, so: Vavg = ½(vi +vf) = (xf – xi )/ t ,or xf = xi + (t/2)(vi +vf)

So far: 1. vf = vi + at 2. xf = xi + (t/2)(vi +vf) Solving 1. for t and inserting into 2 gives you : xf = xi + (vf -vi )(vi +vf)(a/2) Or vf2 = vi 2 +2a(xf –xi)

So far: 1. vf = vi + at 2. xf = xi + (t/2)(vi +vf) 3. vf2 = vi 2 +2a(xf –xi) 2. xf = xi + (t/2)(vi +vf) Substituting 1. into 2. gives: xf = xi + t/2(vi +vi +at), or xf = xi + vit + 1/2at2

Finally, the Four Kinematic Equations 1. vf = vi + at 3. vf2 = vi 2 +2a(xf –xi) 2. xf = xi + (t/2)(vi +vf) 4. xf = xi + vit + (1/2)at2

How To Use Them xf = vf = a = xi = vi = t = 1st List the possible unknowns xf = vf = a = xi = vi = t = 2nd Read the problem and fill in all you can ( usually 4 ). 3rd Choose a kinematic equation with just one unknown in it.

Example xf = vf = a = xf = ? vf = 40 a = ? xi = vi = t = A car starts from rest and accelerates to 40 m/s in 10 seconds. 1st List the possible unknowns – fill in. xf = vf = a = xi = vi = t = xf = ? vf = 40 a = ? xi = 0 vi = 0 t = 10 What is the acceleration of the car? How far does the car move?

To get a, you need an equation with a in it, but without xf To get a, you need an equation with a in it, but without xf. Which one is it? vf = vi + at xf = ? vf = 40 a = ? xi = 0 vi = 0 t =10 40 = 0 + 10a or a = 4.00 m/s2

To get xf you have a choice. Using 2., xf = xi + (t/2)(vi +vf) xf = ? vf = 40 a = ? xi = 0 vi = 0 t =10 xf = 0 + (10/2)(0 +40) = 5(40) = 200m

Example xf = 100 vf = 0 a = ? xf = vf = a = xi = 0 vi = 50 t = ? A car moving at 50m/s sees a dog in the road 100m ahead. If the drivers stops just in time, what acceleration must the brakes provide? How long does it take to stop? xf = 100 vf = 0 a = ? xi = 0 vi = 50 t = ? xf = vf = a = xi = vi = t = What is the acceleration of the car? What is t?

vf2 = vi 2 +2a(xf –xi) 0 =502 + 200a or a = -502 / 200 = -12.5 m/s2 To get a, you need an equation with a in it, but without t. Which one is it? vf2 = vi 2 +2a(xf –xi) xf = 100 vf = 0 a = ? xi = 0 vi = 50 t = ? 0 =502 + 200a or a = -502 / 200 = -12.5 m/s2

To get t, you have a choice. Let’s use: vf = vi + at xf = 100 vf = 0 a = ? xi = 0 vi = 50 t = ? 0 =50 -12.5t or t = -50 /-12.5 = 4.00 seconds

Summary 1. vf = vi + at 2. xf = xi + (t/2)(vi +vf) 3. vf2 = vi 2 +2a(xf –xi) 2. xf = xi + (t/2)(vi +vf) 4. xf = xi + vit + (1/2)at2

EXIT A relay runner moving at 2 m/s, speeds up to 6m/s in 4 seconds. What is her acceleration?