Problem 13 - Calculus

Question

A car is driven at an increasing speed. Sketch a graph of the distance the car has traveled as a function of time.

Step-by-Step Solution

Answer

The graph is a curve that starts at the origin and becomes steeper with time.

Step 1: Understand the Relationship

The car is driven at an increasing speed, meaning its speed is accelerating. This implies that the distance traveled over equal time intervals will increase.

Step 2: Set Axes for the Graph

Set up a coordinate system for the graph: The horizontal axis (x-axis) represents time, and the vertical axis (y-axis) represents distance.

Step 3: Plot the Initial Point

When time is zero (at the origin), the car has not traveled any distance yet. Therefore, the starting point of the graph is (0, 0).

Step 4: Draw the Curve

Because the speed is increasing, the graph should be a curve that starts at the origin and becomes steeper as time progresses, indicating that the car covers more distance with each passing moment. This is typically a quadratic curve, showing constant acceleration, such as an upward-opening parabola.

Key Concepts

Coordinate System

When sketching a graph for any mathematical problem, setting up a coordinate system is the vital first step. A coordinate system allows us to represent values and their relationships visually on a plane, which makes it easier to analyze and understand the problem. Consider it as creating a map where the horizontal component is time and the vertical component is distance in our specific exercise.

• The horizontal axis (also known as the x-axis) represents time. It's usually placed at the bottom of the graph, extending from the left to the right.
• The vertical axis (known as the y-axis) represents distance in this context. It extends from the bottom of the graph upwards.

Setting these axes allows you to plot points that correspond to the car's distance at various moments in time. Initially, when time is zero, the car has not traveled any distance, so your starting point, known as the origin, will be at the coordinates (0, 0). As time progresses, points will be plotted where the longer times correlate with greater distances, showing how the car's travel progresses over time.

Quadratic Curve

In our scenario, where a car is driven with increasing speed, the graph representing the relationship between distance and time will form a quadratic curve. But what is a quadratic curve? Simply put, it is a parabola – a curved line that can open upwards or downwards. In this context, because the speed increases constantly, the curve opens upwards, representing the exponential growth of distance relative to time.

• Quadratic curves are described by a polynomial equation of degree 2, taking the general form: \( y = ax^2 + bx + c \).
• As the vehicle's speed increases at a steady rate (meaning constant acceleration), the curve starts at the origin and becomes steeper over time.

This means each additional unit of time contributes to a more significant increase in distance compared to the previous unit, manifesting as a steeper section of the curve. Understanding the nature of quadratic curves can help us visualize how rapidly acceleration affects distance over time.

Constant Acceleration

Constant acceleration refers to a scenario where an object increases its speed by a fixed amount within every unit of time. This is an essential concept for understanding motion in physics, especially when visualizing distance and time on a graph.

• With constant acceleration, each equal time interval sees an equal increase in velocity, contributing ever-increasing amounts to the total distance traveled.
• This steady increase in velocity results in a quadratic relationship on the graph. Hence, the distance-time graph takes the form of a parabola.

In simpler terms, if a car accelerates constantly, not only does it cover more distance over each time unit, but the rate at which it covers additional distance also escalates. This is precisely why, on the graph, you see a curve whose steepness grows, demonstrating how distance rises more rapidly with constant acceleration.