The "Unreal" Math Behind Catching A Bus

8 min read
The "Unreal" Math Behind Catching A Bus

One of the classic applications of equations of motion while studying Newtonian mechanics is to find the time required by a person running after an accelerating bus to catch. The algebra involved in this problem is not too hard and there is quite instructive.

Such problems contain various physical parameters like distance, speed, acceleration and time. The parameters which are physically measurable are usually denoted by real numbers.

It is believed that complex numbers cannot directly be associated with quantities that can be measured in the real world. If there is ever a scenario where these parameters somehow assume complex values, we would have to rethink about it. In a nutshell - Complex valued parameters are deliberately avoided from Newtonian mechanics.

In this article we are going to see an interesting fact about the possibility of representing one of these physical parameters with complex numbers and its interpretation.

How to catch a bus?

Suppose there is a bus standing at a bus stop. A person is standing at some distance away from the bus. Suddenly, the bus starts to accelerate, and the person starts to run immediately towards it. Our task is to find the time required to catch the bus, if the person actually manages to catch it.

We assume the following -

  • The initial distance between the bus and the person is $d$.
  • The bus starts from rest, at a constant acceleration of $a$.
  • The person starts running towards the bus with constant speed $v$.

To solve the problem, we would require the following equations of motion -

  • For an accelerating body: The distance $s$ covered in time $t$ by a body moving with initial velocity $u$ and constant acceleration $a$ is given by the equation

$$s = ut + \frac{1}{2}at^2$$

  • For a non-accelerating body: The value of $a$ in the previous equation is zero. So the distance $s$ covered in time $t$ is given by
    $$s = ut$$
    These equations can be found in any introductory textbook on Newtonian mechanics.

Let us have a look at a diagram to understand the situation.

Suppose that the person is able to catch the bus in time $t$. Now, what do we mean by catching the bus? - It means that the person is able to run faster than the bus and cover the same distance as that of the bus in time $t$, and they meet at a point. Clearly, the bus is ahead of the person, so an extra distance $d$ has be to covered.

First, we calculate the total distance covered by the bus in time $t$, using the first equation of motion. Since we are given that the bus starts from rest, the initial velocity $u$ of the bus is zero, i.e. $u=0$. So, we get the following result
$$d_{bus} = 0 + \frac{1}{2}at^2$$

Next, we calculate the distance covered by the person in the same time $t$, using the second equation of motion discussed earlier. The result is
$$d_{person} = ut$$
But, we know that an additional distance of $d$ must be covered along with the distance covered by the bus. So we should write this as
$$d_{person} = d + d_{bus}$$

Hopefully, this last equality makes sense. If not, take some time to re-think before proceeding.

By combining these three results into one equation, we get
$$ut = d + \frac{1}{2}at^2.$$

We can rearrange it to get a more familiar form
$$\frac{1}{2}at^2 - ut + d = 0.$$

This is a simple quadratic equation in $t$, and its has two solutions. The methods of finding those solutions are well known. For this particular equation, the two solutions are given by
$$t = \frac{u \pm \sqrt{u^2 - 2ad}}{a}$$

These solutions will tell us the time taken by the person to catch that bus.

When Time Is "Real"

From elementary mathematics, it is known that the square root of a positive real number is always a real number. Since we have a square root in our solutions, and we are interested in real values, the quantity inside the square root must be greater than or equal to zero. That means
$$u^2 - 2ad \ge 0 ~~~ \Longrightarrow ~~~ d \le \frac{u^2}{2a}$$

This relation is telling us that given the values of the speed of the person, $u$ and the acceleration of the bus, $a$, the distance between the bus and the person must be less than $u^2/2a$; then and only then it is possible to catch the bus in time!

This makes sense, as the distance $d$ should not be too large to make it impossible to catch the bus. This condition has another interpretation - If the quantity inside the square root is negative, we will not get a real number as the result (more on this later). If we cannot obtain a real value of time, then that case is useless and we can ignore it. This is the mathematical version of saying that the distance $d$ is too large for us to catch the bus, for a given set of parameters.

Alright, so we know we have to deal with real numbers, but the story of chasing the bus is still not over. We can see that we have two solutions in general, for the value of time $t$. Let us write them separately -

  1. The first solution is
    $$t_1 = \frac{u + \sqrt{u^2-2ad}}{a}.$$

  2. The second solution is
    $$t_2 = \frac{u - \sqrt{u^2-2ad}}{a}.$$

Both of these solutions are mathematically valid. But, that sounds absurd - If we are running along a straight line, how come we have two values of time durations to reach the same spot ?

From the expressions above, we can see that the numerator of $t_1$ has an addition of two positive quantities, while in $t_2$ there is a subtraction. So, we can deduce the following
$$\begin{aligned}\text{If } u^2 > 2ad ~~~ \Longrightarrow ~~~ t_1 > t_2 \\ \text{If } u^2 = 2ad ~~~ \Longrightarrow ~~~ t_1 = t_2 \end{aligned}$$

That means, $t_1$ is either equal to $t_2$ in which case there is no issue of multiple values of time; or $t_1$ has the smaller value of time duration. We expect that if there are two values of time to reach the same spot, the one which makes us go faster makes more sense. Once we are there, we are done; maybe the larger value of time is just a mathematical artifact, with no actual meaning.

It turns out that the other value of time is also meaningful, but in a very unusual way. We obviously know that if the person has to catch the bus, he/she needs to run faster than the bus; there is no way around it. Now, there are two possibilities - The distance $d$ could be such that, the person may run just fast enough to catch the bus and that's the end of it. Another possibility is if the person is running too fast, and instead of catching, he/she overtakes the bus and keeps on running with the same constant speed, leaving the bus behind. It sounds funny, right ?

Now, the bus is behind the person, and is accelerating (gaining speed). It will eventually meet the person at a later time. It is this value of time that $t_2$ represents, and obviously it has to be greater than $t_1$. So we see that both the solutions to our quadratic equation make sense.

When Time is "Unreal"

In the previous section, we discussed in detail, the solutions where time was represented by real numbers. We just had to be careful about the square roots of negative numbers, and the rest is fine.

It turns out that we can use negative numbers inside square roots, leading to superficial values of time, and still make sense out of it!

Alright, so the square root of a negative number is not real; such numbers are called complex numbers. We are not going to discuss much about complex numbers here and we are directly going to use some results for our case.

Let us begin by re-writing the solutions of the quadratic equation as
$$t = \frac{u \pm \sqrt{u^2 - 2ad}}{a}.$$

Now we assume that the quantity inside the square roots is negative, which means
$$u^2 - 2ad < 0 ~~~ \Longrightarrow ~~~ 2ad > u^2.$$

We can re-arrange our expression in the following manner
$$t = \frac{u \pm \sqrt{-(2ad - u^2)}}{a}$$

Now, we use the following mathematical fact from complex numbers -
$$\sqrt{-n} = i\sqrt{n}$$

where $i$ is called "iota", which represents that the square root of $-n$ is an imaginary number, and $n$ is a positive real number whose square root is real as usual.

This means, we can write our solutions in the following way
$$t = \frac{u \pm \sqrt{-(2ad - u^2)}}{a} = \frac{u \pm i \sqrt{(2ad - u^2)}}{a}$$

Since the quantity inside the square roots in now a positive real number, we are fine. Any quantity which is written in the form $a+ib$, is called a complex number. Here $a$ is called the real part, and $b$ is called the imaginary part. Therefore, we have represented the time parameter as a complex number.

Now, we need to interpret this result. We can imagine that, as the person is running after the accelerating bus, the distance between them gets reduced over time. This situation can be visualized as follows

By using the equations of motions discussed earlier, and basic geometry of line segments, we can see from the diagram that the separation between the person and the bus after time $t$ is given by
$$x = d - ut + \frac{1}{2}at^2$$

It's easy to see that this equation makes sense, because when the separation becomes zero, i.e. $x=0$, we recover the original quadratic equation corresponding to the case when the person reached the bus in time $t$.

Using some algebraic manipulations, we rearrange this equation as follows
$$x = \frac{a}{2}\left( t - \frac{u}{a}\right)^2 + \left(d - \frac{u^2}{2a}\right)$$

Upon simplifications, one can easily check that it's the same equation as above. For a given set of values of $d, u$ and $a$, the second term in the brackets does not change over time. The first term in the equation contains the square of a quantity, that means its value can never be negative; its least possible value is zero. This means that the minimum separation between the person and the bus corresponds to the mathematical condition
$$\left(t - \frac{u}{a}\right)^2 = 0 ~~~ \Longrightarrow ~~~ t = \frac{u}{a}$$

This is the value of time duration after which the person will be closest to the bus. But how is this useful? Well, let us rewrite the complex representation of time $t$ that we obtained earlier -
$$t = \frac{u \pm i \sqrt{(2ad - u^2)}}{a} = \frac{u}{a} \pm \frac{ i \sqrt{(2ad - u^2)}}{a}$$

If we look closely, we can see that the real part of $t$ is exactly the same term that we just obtained for least separation!

That means, we can represent the time parameter with a complex number, as long as we only take the real part, with the interpretation that it is the time duration for minimum separation between the person and the bus. Although, we obviously cannot catch the bus, but we can get as close as possible in time $t = u/a$. We can find the value of this minimum separation by plugging this value of $t$ in the expression of $x$ to get
$$x = d - u \cdot\frac{u}{a} + \frac{1}{2} a \cdot \frac{u^2}{a^2} = d - \frac{u^2}{2a}$$

So, we have discovered that sometimes, a complex valued representation of measurable parameters can reveal interesting and useful ideas.

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