Problem 26 - Calculus Early Transcendentals: Pearson New International Edition

Question

From what height must a ball be dropped in order to strike the ground with a velocity of \(-136\) feet per second?

Step-by-Step Solution

Answer

The ball must be dropped from a height of 289 feet.

Step 1: Understanding the Problem

To solve this problem, we will use the equation of motion for a freely falling object. The final velocity of an object in freefall is given by the equation: \( v = \sqrt{2gh} \), where \( v \) is the final velocity, \( g \) is the acceleration due to gravity (32 ft/s² for feet per second), and \( h \) is the height. We need to find the value of \( h \) when \( v = 136 \) ft/s.

Step 2: Rearranging the Equation

We start with the equation \( v = \sqrt{2gh} \). To find \( h \), square both sides to eliminate the square root: \( v^2 = 2gh \). Now solve for \( h \) by dividing both sides by \( 2g \): \( h = \frac{v^2}{2g} \).

Step 3: Substituting Known Values

Substitute \( v = 136 \) ft/s and \( g = 32 \) ft/s² into the equation \( h = \frac{v^2}{2g} \). This results in: \( h = \frac{136^2}{2 \times 32} \).

Step 4: Calculating the Height

Calculate \( 136^2 \): \( 136 \times 136 = 18496 \). Now divide by \( 64 \): \( h = \frac{18496}{64} = 289 \).

Step 5: Conclusion

The ball must be dropped from a height of 289 feet to strike the ground with a velocity of 136 feet per second.

Key Concepts

Freefall

Freefall is a unique type of motion where an object moves solely under the influence of gravity. This means no other forces, such as air resistance, are acting on the object. When we consider an object in freefall:

• The initial velocity is often assumed to be zero unless stated otherwise.
• The object accelerates downward due to the force of gravity.

The beauty of freefall lies in its simplicity. Only one predominant force is acting, which simplifies several calculations in classical physics. Since air resistance is typically ignored in basic physics calculations, freefall equations are especially useful for understanding idealized motion scenarios. Although in reality, objects falling from significant heights would encounter air drag, scientists and educators commonly use freefall conditions because they offer a clearer view of fundamental dynamics.

Acceleration Due to Gravity

Acceleration due to gravity is a crucial constant in physics, denoted as "g". On Earth, this value is approximately 9.8 m/s² or 32 ft/s². Regardless of the mass of the object, in a vacuum, all objects accelerate towards the Earth at this rate when in freefall. Some vital points about gravity include:

• On Earth, "g" is relatively constant, but it can vary slightly depending on location.
• "g" is a vector quantity, meaning it has both magnitude and direction, pointing straight down towards the center of the Earth.

In the context of the original problem, understanding the acceleration due to gravity is vital for using the equations of motion that allow us to solve for height or velocity during freefall. Gravitational acceleration helps us link potential energy changes to kinetic energy in freely falling objects, making it a cornerstone concept for solving motion problems.

Velocity Calculation

Calculating velocity in the context of freefall involves understanding the relationship between displacement, acceleration, and final velocity. For a freely falling object, several equations can help determine its velocity at any given point:

• One fundamental equation is the kinematic equation \( v = \sqrt{2gh} \).
• This formula directly correlates the final velocity \( v \) to the height \( h \) from which the object falls, under constant acceleration \( g \).

In our specific problem, we used this equation to back-solve for the height given a known final velocity. By rearranging the formula to \( h = \frac{v^2}{2g} \), we can calculate the required height for the desired impact velocity. This highlights how motion equations can adapt depending on known and unknown variables in a physics problem, enabling precise calculations of an object's displacement or speed in ideal conditions.