Problem 52 - Calculus with Concepts in Calculus

Question

Let \(h(t)\) denote the height at time \(t\) of an object moving in a vertical direction above the ground. Assume that the object is subject only to the force of gravity. a. Show that the graph of \(h\) is concave downward. b. In terms of the behavior of the velocity \(v\), tell why the result of (a) is true.

Step-by-Step Solution

Answer

The graph is concave downward because the second derivative is negative, and velocity decreases over time due to gravity.

Step 1: Express the height function

The height function for an object under the influence of gravity can be given by the equation: \[ h(t) = h_0 + v_0 t - \frac{1}{2} g t^2 \] where \( h_0 \) is the initial height, \( v_0 \) is the initial velocity, and \( g \) is the acceleration due to gravity (typically 9.8 m/s²). This is a quadratic function of time \( t \).

Step 2: Find the second derivative of the height function

To show concavity, we need to find the second derivative of \( h(t) \). First, differentiate \( h(t) \) with respect to \(t\): \[ h'(t) = v_0 - g t \]. Then, take the derivative of \( h'(t) \) to find the second derivative, \( h''(t) \): \[ h''(t) = -g \].

Step 3: Determine the concavity based on the second derivative

The second derivative \( h''(t) = -g \) is a constant negative value because \( g \) (acceleration due to gravity) is positive. A negative second derivative indicates that the graph of \( h(t) \) is concave downward.

Step 4: Relate concavity to velocity behavior

The velocity \( v(t) = h'(t) = v_0 - g t \) decreases over time because \( -g \) is negative. As time progresses, the object's upward velocity decreases and eventually becomes negative, indicating a downward movement. This decrease in velocity, represented by the negative slope, aligns with the concave downward shape of \( h(t) \).

Key Concepts

Second Derivative Test

The second derivative test is a useful tool in calculus to determine the concavity of a function. It relies on examining the sign of the second derivative of a function. In simple terms, the second derivative indicates how the rate of change of a function's slope is behaving. If the second derivative is positive, the function is concave upward, resembling a cup that can hold water. Conversely, if the second derivative is negative, the function is concave downward, like a frown.

Let's break it down further:

• If the second derivative, denoted by \( f''(x) \), is greater than 0, the curve is concave upward.
• If \( f''(x) \) is less than 0, the curve is concave downward.
• If \( f''(x) \) equals 0, the test is inconclusive, and other methods may be employed.

In the context of our original exercise, we find that the second derivative of the height function \( h(t) \), which is \( h''(t) = -g \), is negative due to the gravitational pull. This results in the concave downward nature of the motion of the object, as gravity always acts to pull objects toward the Earth's surface.

Quadratic Functions

Quadratic functions, like the one used to model the height of an object in vertical motion, are fundamental in mathematics and physics. These functions are characterized by expressions of the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. Quadratics display a parabolic shape on a graph, either opening upwards or downwards depending on the coefficient of the \( x^2 \) term.

In the equation \( h(t) = h_0 + v_0 t - \frac{1}{2} g t^2 \), this is a quadratic function of time \( t \). The parameter \( -\frac{1}{2} g \) serves as the "\( a \)" in the quadratic formula, determining the direction of the parabola. Since \( -\frac{1}{2} g \) is negative, the graph of \( h(t) \) reveals a parabola that opens downward, visually representing the object's ascent slowing and then descending.

Quadratics are powerful for modeling real-world phenomena:

• They illustrate how initial velocity and gravity impact motion.
• The vertex of a parabola gives the peak of an object's trajectory.

This mathematical model accurately describes the object's path not just in an educational setup but also in real-life situations involving projectile motion.

Velocity and Acceleration Analysis

To understand the behavior of objects in motion, it is critical to analyze both velocity and acceleration. These concepts offer insights into how quickly an object is moving and how its speed changes over time.

The velocity \( v(t) \) is the derivative of the height function \( h(t) \), given by \( v(t) = v_0 - g t \). This expression indicates that velocity starts at an initial value \( v_0 \) and decreases by \( g \) meters per second every second. Since \( g \), the acceleration due to gravity, is a positive constant, the term \( -g t \) implicates a continuous reduction in velocity.

This reduction happens at a linear rate:

• The object rises slower over time until it stops momentarily at the peak.
• Past the peak, it descends, gaining speed downward responsibly to gravity.

Acceleration, being the derivative of velocity, is simply \( -g \), a constant negative value, confirming the consistent downward force of gravity. This negative acceleration ensures that velocity decreases, which aligns perfectly with the graph's concave downward nature, conclusively supporting the object's motion in the exercise context.