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.

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 y-axes, m is the gradient and c the intercept on the y axis.

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.




The straight-line 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 y-axis gives the initial speed.

With the set up shown in Fig.3 the times to accelerate between initial and final speeds can be found.

In Fig.2, the fact that the acceleration is uniform between speeds u and v allows us to state that the average speed is




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 y-axis and time squared on the x-axis. The gradient of the line will then be ½a, so the value of the acceleration can be determined.

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 t² 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 second equation of motion states that
 


and the first states that


 



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.

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.




For a constant acceleration, a graph of v² 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 v² 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