Problem 34 - The Calculus with Analytic Geometry

Question

A stone is thrown downward with an initial velocity of \(v_{0} \mathrm{ft} / \mathrm{sec}\). Neglect air resistance. (a) Show that if \(v \mathrm{ft} / \mathrm{sec}\) is the velocity of the stone after falling \(s \mathrm{ft}\), then \(v=\sqrt{v_{0}{\underline{\phantom{xx}}}^{2}+2 g \mathrm{~s}}\). (b) Find the average velocity during the first \(100 \mathrm{ft}\) of fall if the initial yelocity is \(60 \mathrm{ft} / \mathrm{sec}\). Take \(g=32\) and downward as the positive dinection

Step-by-Step Solution

Answer

The final velocity after 100 ft is 100 ft/sec, and the average velocity is 80 ft/sec.

Step 1: Identify the given variables and equation

The initial velocity of the stone is given as \(v_{0} \mathrm{ft} /\mathrm{sec}\). The acceleration due to gravity is \(g = 32 \mathrm{ft}/\text{sec}^2\). We need to find the velocity \(v\) after the stone has fallen a distance \(s \mathrm{ft}\).

Step 2: Apply the kinematic equation

Use the kinematic equation \[v^2 = v_{0}^2 + 2 g s\]. This equation relates the initial velocity, the acceleration, and the distance fallen to the final velocity.

Step 3: Solve for the final velocity

Take the square root of both sides of the equation from step 2 to solve for \(v\): \[v = \sqrt{v_{0}^2 + 2 g s}\]. This confirms the given relationship.

Step 4: Calculate the final velocity after 100 ft

Given the initial velocity \(v_{0} = 60 \mathrm{ft}/\text{sec}\) and \(s = 100 \mathrm{ft}\), substitute these values into the equation: \[v = \sqrt{60^2 + 2 \cdot 32 \cdot 100}\] \[v = \sqrt{3600 + 6400}\] \[v = \sqrt{10000}\] \[v = 100 \mathrm{ft}/\text{sec}\].

Step 5: Find the average velocity

To calculate the average velocity during the first 100 ft of fall, use the formula for average velocity: \[\text{Average Velocity} = \frac{(v_{0} + v)}{2}\]. Substitute the values, \[\text{Average Velocity} = \frac{(60 + 100)}{2} = \frac{160}{2} = 80 \mathrm{ft}/\text{sec}\].

Key Concepts

Initial Velocity

When an object starts its motion, the speed at which it begins is called the initial velocity, often denoted by \( v_0 \). In our example, a stone is thrown downward with an initial velocity. This means at the moment of release, its speed is predetermined:

• The initial velocity in our problem is given as \( 60 \mathrm{ft/\ sec}\).
• Initial velocity sets the stage for all subsequent motion of the object.

Understanding the initial velocity helps us calculate various other aspects of the object's motion, such as its future velocity, distance covered, and time of travel.

Acceleration Due to Gravity

Acceleration due to gravity, represented by \( g \), is the rate at which an object accelerates when it is in free fall toward the Earth. The value of \( g \) is approximately \( 32 \mathrm{ft/s^2} \) (or \( 9.8 \mathrm{m/s^2} \) in metric units):

• Gravity acts downward, consistent with the positive direction assumed in our problem.
• It continuously increases the velocity of the falling object.

This acceleration is crucial for predicting how the velocity of our stone changes over time. By adding the impact of gravity, we get a better understanding of the stone's actual motion.

Average Velocity

Average velocity is the total displacement divided by the total time taken. For a uniformly accelerating object, average velocity can also be calculated using the initial and final velocities:

• Formula: \ \text{Average Velocity} = \frac{( v_0 + v )}{2} \
• In our example: \ \[ \text{Average Velocity} = \frac{ ( 60 + 100 ) }{2\} \= \80 \mathrm{ft/s} \ \]

It's a way to find a middle-ground speed for the object's journey over a specified distance or time. It's particularly useful in problems involving constant acceleration.

Final Velocity

The final velocity of an object in motion is the speed of the object after it has traveled a certain distance or for a set amount of time. In our problem, we use the kinematic equation to find the final velocity \( v \):

• Final velocity after falling \100\ ft\: \ \[ \ v\=\sqrt{60^2\ttt+2\tt 32 \times\100 \}\tt=100\ft/s\ \ \]
• The formula \[ v\=\sqrt{ v_0^2\ttt+2\g\ttt \tt \]} \ allows us to find the final speed given the initial conditions and the distance fallen.

Knowing the final velocity is essential for understanding the dynamics of the object's fall and is used to compute other important values like average velocity.