This post will cover Unit 1, Module 1, Section 3, Parts 1-7.
First, let's look at a couple terms necessary to know before we tackle motion.
Displacement- Displacement is a vector quantity representing the magnitude (size) and direction of the shortest path between two points.
Remember: Vector quantities have both magnitude and direction, while scalar quantities only have magnitudes.
As you can see, displacement is simply the straight line directional space between A and B, while distance, a scalar quantity, encompasses all of the contradictory turns taken between the two points, regardless of whether the net movement is the same or the direction of the movement.
Speed- Speed is a scalar quantity representing the distance travelled per unit time (SI Unit m/s).
Instantaneous vs Average Speed
If you check the odometer of a vehicle in motion, you will se the instantaneous speed, or the value of speed at that exact point in time (this can be determined as the gradient of a distance-time graph at a specific point). However, if you were to measure the total distance travelled by a vehicle over a journey and then divide by the total time taken to complete the journey, the result would be the average speed of the vehicle throughout the journey.
Velocity- Velocity is a vector quantity defined as the displacement per unit time (SI Unit also m/s, but as a vector quantity, it has a direction).
Acceleration- Acceleration is the change in velocity per unit time (SI Unit ms^-2). Note that it is a vector quantity, but we are usually only concerned with the magnitude of acceleration.
Acceleration can be determined as:
Where v is final velocity, u is initial velocity and t is the time throughout which the acceleration (or deceleration occurs).
Graphing Displacement, Speed, Velocity, and Acceleration
Most frequently when it comes to motion, you'll see graphs of one of the above quantities versus time. These graphs are useful for determining the values of the other quantities.
Note: These graphs will only represent motion in a single dimension (i.e. one dimensional motion).
The velocity can be determined as the gradient of a displacement-time graph.
The total displacement will just be the final value of displacement on the graph (y-value of the last point on the graph).
The graph above shows a body travelling at a constant velocity (the quantity is velocity since the graph is displacement versus time rather than distance versus time).
The graph above shows a body at rest (velocity of zero). Consequently, the gradient of the graph is also zero.
This graph shows a body accelerating uniformly. The gradient at a certain point of the graph will give the velocity at the specific time.
The acceleration of an object can be determined as the gradient of a velocity-time graph.
The total displacement can be calculated as the area under a velocity-time graph. For example, the graph below forms a triangle with the x-axis, meaning that the area beneath is could be calculated as bh/2 (b-base, h-height). The base would be the total time while height would be final velocity. Other graphs may form trapezoids or more complex shapes, requiring other formulae as needed.
The graph above shows an object accelerating uniformly from rest.
This graph is of an object decelerating uniformly.
This graph is of an object moving at uniform velocity.
This graph is of an object whose acceleration is increasing from rest.
The velocity of an object will be the area under an acceleration time graph. It's unlikely that you'll encounter acceleration time graphs of gradients that aren't zero.
The graph above shows an object accelerating uniformly.
The Kinematic Equations (and their Derivations)
Soon, we'll meet 5 of your best friends for the entire topic of motion- the kinematic equations! (You now have permission to celebrate). First, let's think about an arbitrary velocity time graph, showing an object which accelerates from an initial velocity (known as u) to a final velocity (known as v) over a time (t):
The graph takes the form of a trapezium. To determine the displacement of the object, we simply find the area under the graph. Since the area of a trapezium is found as:
And the Area (A) is the displacement (s), a can be said to be u, b can be v, and h can be t:
And there's your first kinematic equation! We'll use this to derive all the others, along with the equation for acceleration, which, if you remember, is:
If we transpose the equation for v (i.e. make v the subject by performing operations), we'll see another kinematic equation:
By substituting the second equation into the first, we can determine our next kinematic equation:
The final kinematic equation can be derived by rearranging a previous equation for the value of t, the substituting this value of t into the equation above:
Then, multiplying through by a:
Dividing by a half (multiplying by two) allows us to get rid of the final fraction:
And there is the final horseman of the kinematic apocalypse.
(Note: These equations can only be used to solve problems in which acceleration is uniform or constant)
In the forms given above, these equations are used in the solution of linear motion problems (movement of objects in a straight line), for example, a ball falling in a straight line, or any vehicle travelling in a straight path.
Note: the value of acceleration in your linear motion calculations will be reliant on whether the motion is vertical (affected by acceleration to gravity, 9.81 m/s^2) or horizontal (not affected by gravity).