# Kinematics

#### Displacement, speed, velocity and acceleration:

Distance: Total length covered __irrespective of the direction of motion__.

Displacement: Distance __moved in a certain direction__.

Speed: Distance travelled per unit time.

Velocity: is defined as the rate of change of displacement, or, displacement per unit time

{**NOT**: displacement __ over__ time, nor, displacement

**, nor, rate of change of displacement per unit time}**

__per second__Acceleration: is defined as the rate of change of velocity.

#### Using graphs to find displacement, velocity and acceleration:

- The area under a velocity-time graph is the change in displacement.
- The gradient of a displacement-time graph is the {instantaneous} velocity.
- The gradient of a velocity-time graph is the acceleration.

#### The 'SUVAT' Equations of Motion

The most important word for this chapter is SUVAT, which stands for:

- S (displacement),
- U (initial velocity),
- V (final velocity),
- A (acceleration) and
- T (time)

of a particle that is in motion.

Below is a list of the equations you MUST memorise, even if they are in the formula book, memorise them anyway, to ensure you can implement them quickly.

1. | v = u +at | derived from definition of acceleration: a = (v – u) / t |

2. | s = ½ (u + v) t | derived from the area under the v-t graph |

3. | v^{2} = u^{2} + 2as |
derived from equations (1) and (2) |

4. | s = ut + ½at^{2} |
derived from equations (1) and (2) |

These equations apply only if the motion takes place __along a straight line__ and the __acceleration is constant__; {hence, for eg., air resistance must be negligible.}

### Motion of bodies falling in a uniform gravitational field with air resistance:

Consider a body moving in a uniform gravitational field under 2 different conditions:

#### Without Air Resistance:

__ Assuming negligible air resistance__, whether the body is moving up, or at the highest point or moving down, the weight of the body, W, is the

__only force__acting on it, causing it to experience a

__constant acceleration__. Thus, the

__gradient__of the v-t graph is

__constant throughout__its rise and fall. The body is said to undergo free fall.

#### With Air Resistance:

__ If air resistance is NOT negligible__ and if it is projected upwards with the same initial velocity, as the body moves upwards,

__. Thus its speed will decrease at a rate__

**both**air resistance and weight act**downwards**__greater than 9.81 ms__. This causes the

^{-2}**time taken to reach its maximum height reached to be**

__lower__than in the case with no air resistance. The__max height reached__is also reduced. At the highest point, the body is momentarily at rest; __air resistance becomes zero__ and hence the only force acting on it is the weight. The acceleration is thus 9.81 ms^{-2} at this point.

As a body falls, air resistance __opposes__ its weight. The downward acceleration is thus __less__ than 9.81 ms^{-2}. __As air resistance increases with speed__, it eventually equals its weight (but in opposite direction). From then there will be no resultant force acting on the body and it will fall with a __constant speed__, called the * terminal velocity*.

#### Equations for the horizontal and vertical motion:

x direction (horizontal – axis) | y direction (vertical – axis) | |
---|---|---|

s (displacement) | s_{x} = u_{x} ts _{x} = u_{x} t + ½a_{x} t^{2} |
s_{y} = u_{y} t + ½ a_{y} t^{2}(Note: If projectile ends at same level as the start, then s_{y} = 0) |

u (initial velocity) | u_{x} |
u_{y} |

v (final velocity) | v_{x} = u_{x} + a_{x}t(Note: At ma_{x} height, v_{x} = 0) |
v_{y} = u_{y} + atv _{y}^{2} = u_{y}^{2} + 2as_{y} |

a (acceleration) | a_{x}(Note: Exists when a force in x direction present) |
a_{y}(Note: If object is falling, then a_{y} = -g) |

t (time) | t | t |

Parabolic Motion: tan θ = v_{y} / v_{x}

θ: direction of tangential velocity {NOT: tan θ = s_{y} / s_{x} }