Problem 34 - Algebra Form and Function

Question

The velocity of an object tossed up in the air is modeled by the function \(v(t)=48-32 t,\) where \(t\) is measured in seconds, and \(v(t)\) is measured in feet per second. (a) Create a table of values for the function. (b) Graph the function. (c) Explain what the constants 48 and -32 tell you about the velocity. (d) What does a positive velocity indicate? A negative velocity?

Step-by-Step Solution

Answer

Answer: The function \(v(t) = 48-32t\) represents the velocity of an object tossed in the air. The constant 48 signifies the initial velocity of the object at \(t=0\), which is 48 feet per second upwards. The constant -32 represents the acceleration due to gravity acting downwards on the object, causing the velocity to decrease by 32 feet per second for every second the object is in the air.

Step 1: Table of Values

Here is a table of values for \(v(t) = 48-32t.\) | t | v(t) | |------|--------| | 0 | 48 | | 0.5 | 32 | | 1 | 16 | | 1.5 | 0 | | 2 | -16 | | 2.5 | -32 | (b) Graph the function. To graph the function \(v(t) = 48-32t\), we can use the table of values we generated and plot the points on a coordinate system.

Step 2: Plotting Points on The Graph

Using the table of values, plot each point \((t, v(t))\) on the coordinate system and draw a straight line connecting these points. The line will represent the function \(v(t) = 48-32t.\) (c) Explain what the constants 48 and -32 tell you about the velocity.

Step 3: Constant 48

At time \(t=0\), the initial velocity of the object is \(v(0)=48-32(0) = 48\) feet per second. This tells us that the object is thrown upwards at 48 feet per second.

Step 4: Constant -32

The constant -32 represents the acceleration due to gravity, which is acting downwards on the object. It tells us that the velocity of the object will decrease by 32 feet per second for every second the object is in the air. (d) What does a positive velocity indicate? A negative velocity?

Step 5: Positive Velocity

A positive velocity indicates that the object is moving upward. When \(v(t) > 0,\) the object is rising against the force of gravity.

Step 6: Negative Velocity

A negative velocity indicates that the object is moving downward. When \(v(t) < 0,\) the object has reached its highest point and is now falling back to the ground under the influence of gravity.

Key Concepts

Graphing Functions

Graphing functions is a key part of understanding behavior of mathematical models. By plotting the function, you can visually grasp how changes in variables affect the outcome. For the given velocity function, such as \(v(t) = 48 - 32t\), you observe its linear nature since it's in the slope-intercept form. The graph of this function is a straight line. Here's how to graph:

• First, use a table of values to list out points to plot. This includes the time \(t\) and the velocity \(v(t)\).
• Plot these points onto a coordinate system by considering \(t\) as the x-axis and \(v(t)\) as the y-axis.
• Connect these points with a straight line, illustrating how velocity changes over time.

This visual representation helps you see the point where velocity changes from positive to negative, indicating important transitions in motion.

Table of Values

A table of values serves as a numerical snapshot of the function. It provides specific output values \(v(t)\) for select input values \(t\). This is essential in understanding how a function behaves across different moments.For the velocity function \(v(t) = 48 - 32t\), creating a table of values involves:

• Choosing various points in time \(t\), typically spread evenly, such as 0, 0.5, 1, 1.5, etc.
• Calculating \(v(t)\) for each selected \(t\).

With values like: - \(t = 0, v(t) = 48\)- \(t = 1.5, v(t) = 0\)- \(t = 2.5, v(t) = -32\)You can spot trends such as velocities decreasing by 32 feet per second, aligning with gravity's pull.

Acceleration Due to Gravity

Understanding acceleration due to gravity is crucial in physics, especially in projectile motion studies. It's a fundamental force that impacts all objects in motion on Earth.For the function \(v(t) = 48 - 32t\), the number -32 represents the acceleration due to gravity in feet per second squared. Here’s how it functions:

• Every second, the velocity decreases by 32 ft/s because of gravity's constant downward force.
• This rate affects how quickly the object slows down when moving upward and how fast it accelerates when coming back down.

The negative sign is crucial as it signifies the opposing direction of gravitational force compared to upward velocity.

Positive and Negative Velocity

When analyzing motion, understanding positive and negative velocity can clarify concepts of movement direction.

• A positive velocity in the function \(v(t) = 48 - 32t\) indicates an upward movement. For instance, \(v(0) = 48\) ft/s suggests the object is moving away from the earth.
• As time progresses and \(v(t)\) starts decreasing, it will reach zero, marking the peak of elevation.
• After this point, \(v(t)\) becomes negative, signaling that the object is now moving downward or back towards the ground.

This change from positive to negative is a critical phase indicating when the object starts to fall, connected intricately with both the gravitational deceleration and initial launch velocity.