Acceleration

Acceleration describes the change in speed or velocity of an object.  So acceleration can be a vector and a scalar value.  In this course you will deal with mostly vector calculations.  The general formula for acceleration is;

More specifically, when an object is already moving prior to acceleration, the formula can be written as;

where u = initial velocity,  v = final velocity,  t = time taken.


If velocities are stated and shown as a vector diagram, you MUST make sure that you subtract the inititial velocity (shown as Vi in this diagram).  It is the equivalent of -u in the above formula.

Relative Velocity

Because all the examples are vectors please make sure that you state values and coordinates/direction

There are a few ways that we can tackle the problems of relative velocity. The most important thing to do is work out whether or not the observer is stationary or moving.  We will start with a simple example of a stationary observer looking at a plane.

As you can see this is a simple vector addition in one dimension.  Both the plane's engine and the wind contribute to the speed that a stationary observer on the ground may observe.


When a cross wind is the 2nd contributor, a vector addition needs to be done. Make sure your either use Pythagoras or accurate drawing to ensure that you obtain your solution. 

In this case where a boat is crossing a river we are assuming that the observer is on the shore.  This is no different that the above example.  However there is another factor to consider.  Does the current affect the velocity that the boat is travelling across the river?

The answer is no.  Therefore if asked to calculate the time to cross a river you use only the velocity of the boat contributes, not the river velocity and not the resultant velocity of the boat.  This comes up in many questions so be careful.

When considering a moving observer, the calculation is :

V(rel) = V(object) - V(observer)

If the observer is in car A  The the calculation is 40 - 30 = 10ms-1.  That is the observer is moving at 10ms-1 to the right relative to car B.

In this case, lets say the observer is in the red car.  Because the green car is travelling in the opposite direction relative to the observer, and velocity is a vector quantity, we assign it a negative value.  The calculation is below.   

This also works for motion in 2 dimensions as well.  Just be careful to remember it is a vector subtraction not an addition.

Using graphs to interpret motion.

Graphs are a good way to convey descriptions of real world events in a compact form. Graphs of motion come in differing types depending on what they are describing.  They are used with some measurable or calculable value on the y axis (displacement, velocity or acceleration) and time on the x axis.

Lets start with a displacement time graph


Between C & D, the object has not changed is position and so we can assume its not moving.  

Between B & C the displacement away from the start is increasing at a uniform rate so we can assume it is happening at a constant velocity.

Between A & B the displacement away from the start is increasing at a non uniform rate, so we can assume that object is accelerating away.

Between D & E the object is moving again.  But how?  In what direction compared to the line BC?  How do you know this?




Velocity time graphs.



In this case, the object is changing velocity in a positive manner, therefore it is accelerating.  After 10 seconds it moves at a constant velocity for a further 5 seconds.  After 15 seconds it begins to slow until it is stationary at 30 second.  Between 30 and 40 seconds, it is moving with negative velocity, and accelerating backwards compared to the original motion.  After 40 seconds is slows down again and stops at 55 seconds.


Recall that in Maths when graphing:

                    rise
Slope =      _______
                    run


and 



                    displacement
velocity =   ______________
                         time

Therefore the slope of the line can be used to calculate velocity. 

One of the benefits of a velocity time graph is that the area under it is the displacement.  So if you can calculate the areas of triangles, rectangles and squares, you'll be fine at this stage.  Be warned; if you are doing Mathematics or higher, you'll have to calculate the area under curves.




Here are a couple of summing up graphs:
1. Displacement vs time





2. Velocity vs time.

Speed vs velocity, distance vs time, scalars vs vectors.

What's the difference between scalars and vectors?

Put simply:

1. A scalar quantity is represented by a numerical value only.  It is generally representative of the total distance traveled, speeds etc and the direction of these quantities are not needed.
2. A vector quantity is represented by a numerical value and a direction.  It is not concerned with the total distance traveled or changing speeds, merely the shortest straight line value between point A and point B

Distance and Displacement, can these help explain these?

Lets start out with distance.  Distance is the scalar quantity that says, "how far in total did an object move?"  An odometer in a car is a good example of something that measures distance.  This is because it is not concerned with 
the direction of the travel, only the amount of road covered.


Displacement however, refers to how far away in a straight line has an object moved from its starting point.

Using the above example the distance a person has moved is 50 paces.

The persons displacement (shown by the red arrow) is approximately 36 paces in a NE direction. A little knowledge of Pythagoras will go a long way.

What's the difference between scalars and vectors?

Speed is a scalar quantity that explains how fast an object moves or at what rate it travels a measured distance.  Hence

Velocity is a vector quantity which describes the rate at which an object changes position from its start position to its finishing point.  Therefore

eNOTE:  Because displacement notes the direction, velocity must do so as well.



Lets go back to this example and say that the person took 50 seconds to complete their journey.  The speed is 50 paces / 50 seconds, and so is 
1 pace/second.  

The velocity is 36 paces NE / 50 seconds, so the velocity is (approximately) 
0.7 paces/second NE.




Instantaneous speed/velocity and average speed/velocity.

Instantaneous speed/velocity is the speed/velocity at any instant.  It can be measured using a speed gun or a speedometer.

Average speed/velocity is the average of all instantaneous speed over a journey.  This quantity is calculate using the formula above.