Introduction
In earlier units you have been introduced to the concepts of displacement, velocity, and acceleration. Some of these quantities are defined in terms of others, for example,
We can use mathematical equations based on how an object is currently moving to predict its future motion. Mathematical models of the behaviour of moving objects, such as stars and planets, allow predictions to be made about their position and speed at certain times.
The motion of the Earth around the sun is governed by the laws of circular motion but the positions and speeds of objects moving in a straight line can also be modelled mathematically.
Modelling, based on mathematical descriptions of motion and forces, is used extensively to test the viability of engineering structures before construction.
Movement, like that in a roller coaster, usually involves both curved and straight sections. This unit will introduce the mathematical equations governing uniform accelerations or decelerations for objects moving in a straight line.
In earlier units you have been introduced to the concepts of displacement, velocity, and acceleration. Some of these quantities are defined in terms of others, for example,
We can use mathematical equations based on how an object is currently moving to predict its future motion. Mathematical models of the behaviour of moving objects, such as stars and planets, allow predictions to be made about their position and speed at certain times.
The motion of the Earth around the sun is governed by the laws of circular motion but the positions and speeds of objects moving in a straight line can also be modelled mathematically.
Modelling, based on mathematical descriptions of motion and forces, is used extensively to test the viability of engineering structures before construction.
Movement, like that in a roller coaster, usually involves both curved and straight sections. This unit will introduce the mathematical equations governing uniform accelerations or decelerations for objects moving in a straight line.
Information from gradients
Some types of linear motion can be represented by straight lines on appropriate graph axes. The gradients of these straight
lines is easily determined and gives additional information about the motion. The motion represented in Fig.1 does not produce
a straight line graph but the gradient at any specific point can be determined from the tangent.In mathematical notation any straight line has the equation
y = mx + c
where x and y are the quantities plotted on the x and yaxes, m is the gradient and c the intercept on the y axis.
The first equation of motion
An object moving with a constant
acceleration
An object's acceleration is its rate of change of velocity.accelerationa (sometimes called a uniform acceleration) starts with an initial velocity u and achieves a final velocity v in a time of t seconds.The quantities u, v, a, and t are linked mathematically by the equation:
Click to make v the subject of the above equation and note that
This is called the first equation of motion.
If an object starts with a speed u and accelerates in the same direction at a ms^{−2} for t seconds, then it achieves a final speed v ms^{−1}. The motion of this object is represented by the graph shown in Fig.2.
Since the graph shown in Fig.2 is a straight line we can match the equation for the line to the general mathematical straightline equation 
y = mx + c 
To do this we must rearrange v = u + at slightly to show v on the yaxis and t on the xaxis. 
v = at + u 
By comparing the two equations, we can see that the acceleration corresponds to the gradient of the line, and the intercept on the yaxis indicates the initial speed. 
The straightline nature of the graph in Fig.2 can be explained in terms of the first equation of motion, where the gradient of the line indicates the acceleration and the intercept on the yaxis gives the initial speed.
With the set up shown in Fig.3 the times to accelerate between initial and final speeds can be found.
The second equation of motion
In Fig.2, the fact that the acceleration is uniform between speeds u and v allows us to state that the average speed isThe distance s travelled during t seconds can be calculated using 
Substituting our equation for the average speed gives 
We can now substitute v = u + at from the first equation of motion to give 
This is called the second equation of motion. 
The second equation of motion can be expressed as
To display the second equation of motion in the mathematical form of y = mx + c again requires a slight rearrangement.
To plot a graph which will result in a straight line, we must plot distance on the yaxis and time squared on the xaxis. The gradient of the line will then be ½a, so the value of the acceleration can be determined.
Measuring the acceleration due to gravity
Drop the ball in Fig.4 and note how the distance values increase with time as the ball falls.The initial
displacement
An object's displacement quotes both its bearing and distances relative to a fixed reference point. displacement of the ball from the starting position is 0 m. As the ball falls, its displacement increases. The rate at which the ball falls increases, so the graph is a curved line. The shape of the line shows that the speed is increasing
as the ball falls. The ball is accelerating but we cannot as yet determine if the acceleration is constant during the motion.In the analysis above, we have shown that for a constant acceleration a graph of distance s versus time squared t^{2} will produce a straight line. Drop the ball in Fig.5 and step through the stages to complete the table and graph.
The straight line in the graph of Fig.5 confirms that the ball is falling with a constant acceleration. The gradient of the line is ½a and so the actual acceleration of the ball can be found.
A dropped ball accelerates towards the ground because of the gravitational attraction between the earth and the ball. This attractive force acts as an
unbalanced force
Unbalanced forces occur in situations where the force acting in one direction is greater than the force acting in the opposite
direction.unbalanced force which causes an acceleration. When the ball is relatively close to the earth's surface, the force is constant and causes
all objects to accelerate at 9.81 ms^{−2}. This value is often called the acceleration due to gravity g.The third equation of motion
The second equation of motion states that
and the first states that
We can make t the subject of the first equation 
Substituting for t into the second equation of motion gives 
Simplifying and rearranging gives 
By combining the first and second equations of motion we have been able to derive the third:
The third equation of motion is particularly useful in calculations where information about time is not given.
Testing the third equation of motion
If an object starts with a speed u and accelerates in the same direction at a ms^{−2} for t seconds, then it achieves a speed v ms^{−1} and travels a distance s.The third equation of motion links u, v, a, and s and states that
v^{2} = u^{2} + 2as. 
The third equation of motion can be rearranged into the mathematical form y = mx + c 
v^{2} = 2as + u^{2} 
For a constant acceleration, a graph of v^{2} versus s should give a straight line which has a gradient of 2a.
The arrangement shown in Fig.6 uses light gates to determine the speed of a rolling trolley. Place the gate at position A and measure the velocity of the trolley. Reposition the trolley at the top of the runway and move the light gate to position B. Repeat measurements and build up the table of values for the other locations.
When the results table is completed, click through the remaining stages to draw a graph of the results. The straight line in the v^{2} versus s graph confirms that the trolley has a constant acceleration down the slope. The actual acceleration value is half the gradient of the line on the graph.
Summary
Constant velocities and accelerations can be described with equations which also explain the shape of the lines on motion graphs.
The first equation of motion states that
The second equation of motion states that
The third equation of motion states that
Constant velocities and accelerations can be described with equations which also explain the shape of the lines on motion graphs.
The first equation of motion states that
The second equation of motion states that
The third equation of motion states that
Exercises

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