Mechanics

Topic 2.1 Kinematics

Kinematic Concepts:Displacement

Is a measured distance in a given directionIt is a vector quantityIt tells us not only the distance of the object from a particular reference point but also the direction from that reference pointTypically, it is measured from the origin of a Cartesian co-ordinate system

Kinematic Concepts:Speed

Is the rate of change of distanceOr is the distance covered per unit timeIt is a scalar quantity…it has magnitude onlySpeed is the total distance (d) covered in total time (t)Speed (s) = total distance (d)

total time (t)

Kinematic Concepts:Velocity

Is the rate of change of displacementIs a measured speed in a given directionIt is a vector quantity…It tells us not only the speed of the object but also the direction

Average Velocity

Defined as the total displacement (s) of the object in the total time (t)

Velocity (vav) = total displacement (s)total time (t)

vav = s twhere indicates a change in the value

Instantaneous Velocity

Is the velocity at any one instant

v = s t

*Where t is tending towards zero

Kinematic Concepts:Acceleration

Is the rate of change of velocity in a given directiona = v / t (where v = v – u)It is a vector quantityAcceleration in the same direction as motion results in an increase in speedAcceleration in the opposite direction as motion results in a decrease in speedAcceleration perpendicular to the direction of motion results in a change in direction

Graphical Representation of Motion

These come in 5 forms…1. Distance-time graphs2. Displacement-time graphs3. Speed-time graphs4. Velocity-time graphs5. Acceleration-time graphs

Gradiants of Graphs

Gradiant of a Displacement-time graph is the velocity

Gradiant of a Velocity-time graph is the acceleration

Areas Under Graphs

Area under a Velocity-time graph is the displacementArea under a Acceleration-time graph is the velocityAreas can be calculated by the addition of geometric shapes

The Equations of Uniformly Accelerated Motion

There are 4 equations which we use when dealing with constant acceleration problems

You need to be able to derive them

The 4 Equations

Supposing the velocity of a body increases from u to v in time t, then the uniform acceleration, a is given bya = change of velocity

time takena = v – u

t v = u + at - equation (1)

Since the velocity is increasing steadily, the average velocity is the mean of the initial and final velocities, i.e.Average velocity = u + v

2If s is the displacement of the body in time t, then since average velocity = displacement/time = s/tWe can say s = u + v

t 2 s = ½ (u + v) t - equation (2)

But v = u + at s = ½ (u + u + at) t s = ut + ½at2 - equation (3)

If we eliminate t from (3) by substituting in t = (v – u)/a from (1), we get on simplifyingv2 = u2 +2as - equation (4)

Knowing any three of s, u, v, a, t, and the others can be found

Experiments show that at a particular place all bodies falling freely under gravity, in a vacuum or where air resistance is negligible, have the same constant acceleration irrespective of their masses.This acceleration towards the surface of the Earth, known as the acceleration due to gravity, is donated by g.

Acceleration Due to Gravity

Its magnitude varies slightly from place to place on the Earth´s surface and is approximately 9.8ms-2

The IB generally allows for an approximation of 10 ms-1 to be used

All of the uniform acceleration equations are applicable to situations of free fall

The Effects of Air Resistance

Air resistance depends on 2 things Surface area Velocity

Air resistance increases as surface area increasesAir resistance increases as the velocity increases

Terminal Velocity

As an object falls through the air, it accelerates, due to the force of attraction of the Earth. This force does not change.As the velocity increases, the air resistance, the force opposing the motion, increases, therefore the acceleration decreases.

If the object falls for long enough, then the air resistance (a force acting upwards) will equal the force of attraction of the Earth (the weight) (a force acting downwards)Now there are no net forces acting on the object (since the two forces balance) so it no longer accelerates, but travels at a constant velocity called its terminal velocity. A sky diver has a terminal velocity of more than 50ms-1 (100 miles per hour)

Relative Motion

If you are stationary and watching things come towards or away from you, then your stating of velocities is straightforward.If, however you are in motion, either moving towards or away from an object in motion, then your frame of reference is different

In this case the relative velocity is the velocity of the object relative to your motion.Common examples include cars overtaking Trains going passed platforms