An engineer designs a robot that can climb stairs. If the robot climbs with a constant speed v, batt

Q: Why did we assume the graph is a parabola?

The curve is a smooth, symmetric U-shape with a clear minimum on the y-axis, which is the hallmark of a quadratic function $y = ax^2 + c$. In physics problems, such dependencies (like power loss due to drag) are typically quadratic.

Q: Why is the y-intercept not zero?

Even when the robot is stationary ($v=0$), its internal electronics and control systems still consume power, leading to a non-zero baseline battery discharge rate.

Q: Why did we minimize $1/L$ instead of maximizing $L$ directly?

Maximizing $L = \frac{v}{av^2 + c}$ requires calculus (quotient rule). Minimizing $1/L = av + \frac{c}{v}$ allows us to use the much faster and more elegant AM-GM inequality.

Q: How does the AM-GM inequality work here?

The AM-GM inequality states that the sum of two positive numbers is always greater than or equal to twice the square root of their product. Since the product of $av$ and $c/v$ is a constant ($ac$), it perfectly finds the minimum of the sum.

Q: What is the optimal speed for the robot to achieve this maximum distance?

The maximum distance occurs when the two terms in the AM-GM inequality are equal. Setting $0.005v = 0.02/v$ gives $v^2 = 4$, so the optimal speed is $2 \text{ m/s}$.

Question

An engineer designs a robot that can climb stairs. If the robot climbs with a constant speed $v$, battery of the robot discharges completely in a time interval $	au$. This dependence is shown in the following graph. With the help of the graph, determine the maximum length of a staircase, which the robot can climb with a constant velocity.

Sigma Percentile · Accepted Answer

Imagine you are an engineer tasked with designing a robot that can climb stairs. You have a battery, and you know that the faster the robot climbs, the faster the battery drains. But if it climbs too slowly, it might just run out of battery because it takes too long! This is a classic optimization problem, and the graph provided is our map to the solution.

### Analyzing the Setup

Let's start by understanding what the graph is telling us. The horizontal axis represents the constant speed $v$ of the robot in meters per second. The vertical axis represents the rate of battery discharge, given by $1/	au$, where $	au$ is the total time the battery lasts.

Our ultimate goal is to find the maximum length of the staircase, let's call it $L$, that the robot can climb. From basic kinematics, we know that distance is speed multiplied by time. Therefore, the length of the staircase is given by:
$$L = v \cdot 	au$$

However, our graph doesn't give us $	au$ directly; it gives us $1/	au$. So, let's cleverly rearrange our formula to match the information we have:
$$L = \frac{v}{1/	au}$$

To proceed, we need to translate the visual curve on the graph into a rigorous mathematical equation.

### The Master Equation

Look closely at the shape of the curve. It's a beautiful, smooth parabola opening upwards. Even better, its lowest point—the vertex—lies perfectly on the y-axis. In coordinate geometry, a parabola with its vertex on the y-axis has a simple quadratic form:
$$\frac{1}{	au} = a v^2 + c$$

Our mission now is to find the constants $a$ and $c$. We can do this by picking clear, unambiguous points from the grid.

First, let's look at the y-intercept. When the robot is completely stationary ($v = 0$), the discharge rate is not zero! It's $0.02 	ext{ s}^{-1}$. This makes physical sense; even when not moving, the robot's internal electronics consume power. Substituting this into our equation gives us our first constant:
$$c = 0.02$$

Next, we need another point to find $a$. Scanning the grid, we see that at a speed of $v = 2.0 	ext{ m/s}$, the curve perfectly intersects the horizontal grid line at $0.04 	ext{ s}^{-1}$. Let's substitute these values into our growing equation:
$$0.04 = a(2.0)^2 + 0.02$$

Now, it's just a matter of simple algebra. Subtracting $0.02$ from both sides:
$$4a = 0.02$$

Dividing by 4, we find the value of $a$:
$$a = 0.005$$

We have successfully decoded the graph! The master equation governing the robot's battery discharge rate is:
$$\frac{1}{	au} = 0.005 v^2 + 0.02$$

### The Optimization Strategy

Now that we have the discharge rate as a function of speed, let's substitute it back into our length equation:
$$L = \frac{v}{0.005 v^2 + 0.02}$$

We want to find the maximum possible value for $L$. While we could use calculus and take the derivative, there is a much more elegant algebraic trick. Maximizing a fraction with a variable in the numerator is often tricky. But maximizing $L$ is exactly the same as minimizing its reciprocal, $1/L$. Let's flip the equation:
$$\frac{1}{L} = \frac{0.005 v^2 + 0.02}{v}$$

By splitting the fraction, we get:
$$\frac{1}{L} = 0.005 v + \frac{0.02}{v}$$

Look at the structure of this expression! We have the sum of two positive terms, and their product is a constant (because the $v$ in the numerator and denominator will cancel out). This is the perfect scenario to unleash the power of the **AM-GM Inequality** (Arithmetic Mean-Geometric Mean Inequality).

The AM-GM inequality states that for any two positive numbers $x$ and $y$:
$$x + y \ge 2\sqrt{xy}$$

### Final Calculation

Let's apply this powerful theorem to our expression for $1/L$:
$$0.005 v + \frac{0.02}{v} \ge 2\sqrt{0.005 v \cdot \frac{0.02}{v}}$$

Watch the magic happen inside the square root. The variable $v$ completely vanishes!
$$\frac{1}{L} \ge 2\sqrt{0.0001}$$

Since the square root of $0.0001$ is $0.01$, we have:
$$\frac{1}{L} \ge 2(0.01)$$
$$\frac{1}{L} \ge 0.02 	ext{ m}^{-1}$$

We have found that the absolute minimum value for $1/L$ is $0.02$. Therefore, the maximum value for the length $L$ is simply the reciprocal of this minimum:
$$L_{	ext{max}} = \frac{1}{0.02}$$
$$L_{	ext{max}} = \mathbf{50 	ext{ m}}$$

The maximum length of the staircase the robot can climb is 50 meters. This elegant solution highlights the beautiful interplay between graphical analysis, kinematics, and algebraic optimization. By finding the perfect balance between speed and power consumption, we've pushed the robot to its absolute physical limit!

Enter Numerical Value:

Visualized Solution

$Visualizing the Problem$

$Kinematic Relation$

$Rearranging the Formula$

$Equation of a Parabola$

$Finding the y-intercept$

$Using a Data Point$

$Calculating the Constant$

$The Master Equation$

$Optimization Strategy$

$Applying AM-GM Inequality$

$Finding the Minimum$

$Maximum Length$

$Physical Significance$

The Sigma Insight: Motion in a Straight Line

Analyzing the Setup

The Master Equation

The Optimization Strategy

Final Calculation

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Enter Numerical Value:

Visualized Solution

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Visualizing the Problem

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Kinematic Relation

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Rearranging the Formula

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Equation of a Parabola

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Finding the y-intercept

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Using a Data Point

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Calculating the Constant

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The Master Equation

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Optimization Strategy

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Applying AM-GM Inequality

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Finding the Minimum

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Maximum Length

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Physical Significance