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#### Notes by Category University Engineering

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# Acceleration, Projectiles & Kinematics

Velocity is the rate of change of an object's position, and therefore has direction. This makes it a vector quantity, unlike speed which is scalar (magnitude but no direction).

velocity = distance / time

The units of velocity are m/s

Acceleration is the rate of change of velocity, and so is the mathematical derivative of this.

acceleration = change in velocity / change in time

The units of acceleration are m/s²

## Motion Graphs

We can plot motion on two main types of graph - it is important to know the properties of each. ### Displacement-Time Graphs

• The Gradient is the velocity - draw a tangent to find the instantaneous velocity

• Horizontal line represents zero velocity ### Velocity-Time Graphs

• The Gradient is the acceleration

• The Area beneath the graph is the displacement

## Constant Acceleration

When acceleration is constant (e.g. free fall when we ignore air resistance), we can use SUVAT equations to work out the variables: • s is for displacement

• u is for initial velocity

• v is for final velocity

• a is for acceleration

• t is for time

### Vertical Motion due to Gravity

The gravitational force of the earth causes all objects to accelerate towards the ground, its surface. Ignoring air resistance, the acceleration is constant, and given a g:

g = 9.8 m/s²

This is independent of the mass, shape or velocity of the object. It is vital to set a positive direction of motion for each question

In the example above, we set up as the positive direction, so our values for a and v were negative because the ball is going down.

### Constant Acceleration with Vectors

Additionally, we can express motion using vectors:

r = r₀ + v t
• r is the position vector of the moving object

• r₀ is the initial position vector

• v is the velocity vector

Four of the five SUVAT equations have vector equivalents:

• v = u + at v = u + a t

• s = ut + ½at² s = u t + ½ a t² + r₀

• s = vt - ½at² s = v t - ½ a t² + r₀

• s = ½(u+v)t s = ½( u + v )t + r₀

v² = u² + 2as has no vector equivalent

## Projectile Motion

When we model a projectile, we ignore air resistance. This means that: Horizontal motion of a projectile has constant velocity: a = 0
Vertical motion of a projectile motion is modelled as gravitational free fall: a = g

For horizontal projection, like in the diagram above:

• Since horizontal velocity is constant, we can use the equation x = vt (were x is horizontal displacement)

• For vertical velocity, we need to use SUVAT equations, due to the constant acceleration of g = 9.8 m/s²

### Horizontal and Vertical Components

When given the velocity as a vector or at an angle, you must use trigonometry to find the vertical and horizontal components. Then, treat them separately as above (the horizontal component still has constant velocity)