Constant Acceleration Acceleration is how quickly the velocity is changing - the change in velocity per unit time. If the acceleration is constant, this can then be expressed as the following equation: Acceleration = Change in Velocity Time a = (v - u) t Change in velocity = Final velocity - Initial velocity Where: a is acceleration v is the final velocity u is the initial velocity t is time taken

Constant Acceleration From... a = (v - u) t We rearrange the formula to make v the subject... at = v - u (multiply both sides by t) at + u = v (add u to both sides) Rewriting, gives... v = u + at

Calculating the Distance Travelled Consider an object accelerating from an initial velocity u to a final velocity v in t seconds... As before, to calculate the distance travelled, we need to find the area underneath the graph...

Calculating the Distance Travelled at = v - u a = (v - u) From: Area of Lilac Rectangle = Base x Height = t x u = ut Area of Blue Triangle = ½ Base x Height = ½ x t x (v - u) = ½ x t x (at) = ½ at2 s = ut + ½ at2 Total Area = ut + ½ at2 so the formula for distance travelled (s), is…

Average speed = Total distance travelled We also know that the average speed can be calculated from... Average speed = Total distance travelled Total time taken u + v = s t 2 If we multiply both sides of the equation by t we can find out s... s = (u + v) x t 2 This formula allows us to calculate the distance travelled if we know the initial and final velocities and the time taken.

Constant Acceleration Formulas We now have 3 constant acceleration formulas… v = u + at s = ut + ½ at2 s = (u + v) x t 2 We can combine formulas and by eliminating t... The first step is to make t the subject of formula v = u + at v - u = at (subtracting u from both sides) v - u = t a (dividing both sides by a) t = v - u a (rewriting)

Constant Acceleration Formulas The second step is to substitute this value for t into formula s = (u + v) x t 2 t = v - u a s = (u + v) x 2 (v - u) a Multiply both sides by 2a... 2as = (u + v) (v - u) Multiply out the brackets... So the fourth constant acceleration formula is: v2 = u2 + 2as 2as = v2 - u2 Add u2 to both sides and rewrite... v2 = u2 + 2as

Constant Acceleration Formulas - Summary We now have 4 formulas. These formulas will help you calculate any motion problem in which a body undergoes zero or constant acceleration. Symbol Meaning Unit v final velocity m/s u initial velocity s distance travelled m t time taken a acceleration m/s2 Formulas s = (u + v ) x t 2 s = ut + ½at2 v = u + at v2 = u2 + 2as Whenever you have any 3 of the five ‘v-u-s-t-a’ unknowns, you can find out the remaining 2 unknown values by using one or more of the above formulas…

Using Constant Acceleration Formulas s = (u + v ) x t 2 s = ut + ½at2 v = u + at v2 = u2 + 2as

Projectile Motion - Forces Acting Ignoring air resistance, the only force acting on a projectile during the flight is gravity. Projectiles have a downward acceleration (due to gravity) and this only affects the vertical velocity. For a projectile there is no acceleration in the horizontal direction. Horizontal Vertical

Altering Projectile Velocity A cannon ball is fired horizontally at a target. Can you find out (or calculate) the correct velocity to hit the target?

Calculating Time Taken Example: Calculate the time taken, from firing, for the cannon ball to hit the target. t = d/s is a formula that can be applied to solve problems, whenever velocity is constant... Velocity is constant in the horizontal vector... t = d/s t = 48/24 t = 2 Time taken is 2s

Calculating Final Velocity v = u + at is a formula that can be applied to solve problems, whenever acceleration is constant... Example: Calculate the final vertical velocity of the cannon ball as it hits the target. In this case a = g = 9.8ms-2 (9.8m/s2) v = u + at v = 0 + 9.8 x 2 v = 0 + 19.6 v = 19.6m/s Final vertical velocity is 19.6m/s

Kinetic Energy 2 starting equations Force = m x a (Newton’s 2nd law) W = Force x distance Together give Work = (m x a) x distance W = m x a x d

Work is MAD Av Speed (velocity) is distance time Distance = Average velocity x time Distance = v + u x t 2

Distance Distance = v + u x t 2 D = ½ x v + u x t Starting velocity is zero D = ½ x v x t

Time Final velocity = acceleration x time Time = final velocity t = v a

Going back a step….. D = ½ x v x t D = ½ x v x v a D = ½ x v2

Put into previous statements W = m x a x d W = m x a x ½ x v2 a W = m x ½ x v2 W = ½ x m x v2

GPE work done = Force x distance Force = mass x acceleration Gravitational constant (g) is the cause of the acceleration Force = mass x gravitational constant F = m x g Work = mass x acceleration x distance In this case height is the distance therefore, PE=mgh

Work Work = force x distance Measured in Joules. Weight is a force, mass is not

Power Power = work / time (s) Measured in Joules/second or Watts

Newton’s 2nd law Force = mass x acceleration Force in Newton’s Mass in kg Acceleration in m/s2