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- A-Level Maths

# 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:

is for*s***displacement**is for*u***initial velocity**is for*v***final velocity**is for*a***acceleration**is for*t***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+vt

is the position vector of the moving object**r****r₀****v**

Four of the five SUVAT equations have vector equivalents:

v = u + at

__v__=__u__+__a__ts = 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)

For an object projected at velocity U at an angle of θ to the x-axis:

the

**horizontal component**of the velocity is given as**U cos(θ)**the

**vertical component**of the velocity is given as**U sin(θ)**

The vertical component is decelerating at -9.8 m/s² as it rises, and accelerates at 9.8 m/s² as it falls

**The projectile reaches its maximum height when the vertical component of the velocity is zero. **

Applying SUVAT to the vertical component of the projectile gives us a few standard and useful equations:

Projectile motion can also be plotted and calculated with in vectors.

## Variable Acceleration

As you can see from the velocity-time graph, varying acceleration produces curves:

This is because the gradient of a velocity-time graph is the acceleration, and so if the acceleration changes with time, so must the gradient.

### Kinematics

Velocity is the rate of change of displacement. Acceleration is the rate of change of velocity. Therefore:

This means we can differentiate and integrate equations for motion:

### Integrating and Differentiating Vectors

Variable acceleration can also be expressed in vector form. Therefore, you need to be able to differentiate and integrate vector equations.

Often, **dot notation** is used to quickly represent differentiation with respect to time:

To integrate vectors:

In both differentiation and integration of vectors, you must do one term at a time

See Notes Sheets on differentiation and integration in __Pure Maths__ if you need a recap.