Problem 70 - A Graphical Approach to Precalculus with Limits

Question

Solve each problem. A ball thrown straight up into the air has an initial height of 5 feet and an initial velocity of 128 feet per second. What is the velocity of the ball after 2 seconds?

Step-by-Step Solution

Answer

The velocity of the ball after 2 seconds is 64 feet per second.

Step 1: Identify the Given Values

The exercise provides the initial velocity of the ball, which is 128 feet per second. We also know the initial height is 5 feet, but this is not needed directly to find the velocity after 2 seconds. The time elapsed is given as 2 seconds.

Step 2: Use the Formula for Velocity with Gravity

The velocity of an object in free fall (like a ball thrown upwards) can be calculated using the formula: \[ v = v_0 - gt \]where \( v_0 \) is the initial velocity, \( g \) is the acceleration due to gravity (approximately 32 feet per second squared), and \( t \) is the time in seconds.

Step 3: Substitute Values into the Formula

Substitute the known values into the formula: \[ v = 128 - 32 \times 2 \]This takes into account the downward acceleration due to gravity acting on the ball.

Step 4: Calculate the Result

Perform the calculations step-by-step: First, calculate \( 32 \times 2 = 64 \). Then, subtract 64 from the initial velocity: \[ v = 128 - 64 = 64 \]Thus, the velocity of the ball after 2 seconds is 64 feet per second.

Key Concepts

Initial Velocity

In any motion analysis, understanding the initial velocity of an object is crucial. Initial velocity, denoted as \( v_0 \), represents the speed and direction of an object when it starts moving. In our problem, the ball has an initial velocity of 128 feet per second.

Initial velocity:

• Describes how fast and in what direction an object moves at the start.
• Is essential for predicting future motion.

In this context, the ball is launched upwards with an initial boost of 128 feet per second, counteracting gravity momentarily. Knowing \( v_0 \) allows us to predict how the velocity will change over time, influenced by forces like gravity.

Acceleration Due to Gravity

Acceleration due to gravity, represented by \( g \), is the constant rate at which objects accelerate towards the Earth. This value is approximately 32 feet per second squared on Earth.

Key aspects of gravity:

• Acts downwards, always working to slow the upward motion of objects.
• Is consistent, meaning all objects fall at the same rate in the absence of air resistance.

For the ball in our problem, gravity decreases its upward velocity by 32 feet per second every second. This constant force is what allows us to predict how fast the ball will be moving at any given moment during its flight.

Velocity Calculation

Calculating the velocity of an object after a period involves understanding how initial velocity and gravity interact over time. The formula used here is:
\[ v = v_0 - gt \]
Where:

• \( v \) is the final velocity.
• \( v_0 \) is the initial velocity.
• \( g \) is the acceleration due to gravity (32 feet/sec²).
• \( t \) is the time in seconds.

Substituting the known values for our ball:

• Initial velocity \( v_0 = 128 \) feet per second.
• Time \( t = 2 \) seconds.
• The change in velocity due to gravity is \( 32 \times 2 = 64 \).

Subtracting the effect of gravity from the initial velocity gives:\[ v = 128 - 64 = 64 \text{ feet per second} \]Thus, the ball’s velocity after 2 seconds is 64 feet per second, showing how gravity diminishes its upward speed.