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Constant Acceleration

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Change in velocity =

Final velocity - Initial velocity

Acceleration = Change in Velocity

Time

a = (v - u)

t

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:

Where:

a is acceleration

v is the final velocity

u is the initial velocity

t is time taken

From...

a = (v - u)

t

v = u + at

Constant Acceleration

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...

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...

s = ut + ½ at2

Area ofLilacRectangle

Area ofBlueTriangle

From:

= Base x Height

= t x u

= ut

= ½ Base x Height

= ½ x t x (v - u)

= ½ x t x (at)

a = (v - u)

t

at = v - u

= ½ at2

Calculating the Distance Travelled

Total Area = ut + ½ at2 so theformula for distance travelled (s), is…

s = (u + v) x t

Average speed = Total distance travelled

2

Total time taken

u + v = s

2

t

Average Speed

We also know that the average speed can be calculated from...

If we multiply both sides of the equation by t we can find out s...

This formula allows us to calculate the distance travelled if we know the initial and final velocities and the time taken.

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

s = ut + ½ at2

v = u + at

v - u = t

a

t = v - u

a

Constant Acceleration Formulas

We now have 3 constant acceleration formulas…

v = u + at

v - u = at

(subtracting u from both sides)

(dividing both sides by a)

(rewriting)

The second step is to substitute this value for t into formula

s = (u + v) x t

2

s = (u + v) x

(v - u)

2

a

t = v - u

a

So the fourth constant acceleration formula is:

v2 = u2 + 2as

Constant Acceleration Formulas

Multiply both sides by 2a...

2as = (u + v) (v - u)

Multiply out the brackets...

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.

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…

s = (u + v ) x t

2

Using Constant Acceleration Formulas

s = ut + ½at2

v = u + at

v2 = u2 + 2as

Projectile Motion - Forces Acting

Ignoring airresistance, the only force acting on a projectile during the flight is gravity.

Projectiles have a downwardacceleration (due to gravity) and this only affects the verticalvelocity.

For a projectile there is noacceleration in the horizontaldirection.

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

- 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

- Av Speed (velocity) is distance
time

- Distance = Average velocity x time
- Distance = v + u x t
2

- Distance = v + u x t
2

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

- Final velocity = acceleration x time
- Time = final velocity
acceleration

- t = v
a

- D = ½ x v x t
- D = ½ x v x v
a

- D = ½ x v2
a

- 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

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

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

- Force = mass x acceleration
- Force in Newton’s
- Mass in kg
- Acceleration in m/s2