Class 11 Maths: Cricket, Parabolas And Algebra Walk Into The Field

Indians love Cricket. It is even considered a religion by many. If you are a fan of the game, you must know what “Hawk Eye” is. It is a technology that helps umpires predict the path of the ball. Mostly used in the case of Leg Before Wicket (LBW) decisions, it tracks the ball’s path till the point of impact using cameras, and then extrapolates it to the stumps. If the ball is found to be hitting the stumps, below the height of the stumps and in front of them, the batsman is declared out.

Let us understand this process with the help of a mathematical model, for the Hawk Eye’s process is an example of applied mathematics. Let us first quickly recall how to make a straight-line graph.

If we are given the equation of straight line y = x + 1, we can make its graph by putting different values of x in the formula and obtaining corresponding values of y. For example, if we put x=1 in the equation, we get y = 2. So these values of x and y, when put in an ordered pair (1,2), will give us one point (say A) whose coordinates are (1,2) and which lies on the graph. We can similarly put x=2, and obtain y = 3. So, this point B (2,3) also lies on this graph. In the same way, we can obtain more points like C(3,4), D(4,5), etc. Now we can plot these points on the graph paper and join them to obtain the graph of y = x + 1. The graph will be as shown below.

Graph of y=x+1

Now, if we are given the graph, we can also find its equation. If the graph is as shown below, we can select any two points on it, say A (1,3) and B (2,4), and deduce that the equation is y = x +2.

From graph to equation

We also have the method of solving two linear equations to get the exact equation for a complex case but we won’t address that here. The important point to note is that we can make a graph from an equation, and we can also find the equation from the graph by using some computations.

Cricket And Class 11 Maths

Now, we come back to cricket. When the ball is bowled, the path it follows is a parabola. Parabola is a curve which can be represented by the general expression where a, b, c are constants, which are different for different parabolas (this is a simplified form of parabola). So, if a computer can plot the path of the ball on graph paper, and compute the equation from the graph, it can predict whether the ball will hit the stump or not.

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The following exercise will make it clearer. Say, after the ball bounces off the pitch, computers plot the path of the ball till it hits the batsman’s pads. This path will be a parabola. If the equation of the path calculated by the computer is can we tell conclusively if the ball hit the stumps? Let’s find out.

Predicting the path of the ball

As it is clear from the graph, the value of x at the stumps is x = 2, and the height of the stumps is 1 (units are in meters). Let us find the height (y coordinate) of the ball, when it reaches the stumps. Putting x = 2 in the equation, meters. Clearly, the ball’s height is more than the stumps, and it will not hit the stumps, and thus, the batsman is not out.

We can similarly have different instances with different equations and check whether the ball would have hit the stumps or not. Now, let us make you the umpire and let you find out whether a batsman is out or not for the following scenarios:

1. The stumps are at x = 3, and the equation of the line after pitching is

The height of the stumps is 1 meter.

2. The stumps are at x = 5, and the equation of the line after pitching is

The height of the stumps is again 1 meter.

Through these exercises, we hope you learned about the relationship between graphs and equations. In your studies, you should try to think of different functions in terms of graphs. That makes some problems, which otherwise look very complex using pure algebra, much easier to visualize and solve.