Problem 33 - The Calculus with Analytic Geometry

Question

A ball is dropped from rest and after \(t\) sec its velocity is \(v \mathrm{ft} / \mathrm{sec}\). Neglecting air resistance, show that the average velocity during the first \(\frac{1}{2} T\) sec is one-third of the average velocity during the next \(\frac{1}{2} T \mathrm{sec}\)

Step-by-Step Solution

Answer

The average velocity during the first half is one-third of that during the second half.

Step 1: Understand the given problem

The problem states that the ball is dropped from rest, meaning the initial velocity is zero. The exercise asks to show that the average velocity during the first half of the total time period is one-third of that during the second half, assuming no air resistance.

Step 2: Express velocity using kinematic equations

For an object in free fall, the velocity at time t can be expressed as: \( v = g t \) where \( g \) is the acceleration due to gravity (approx. 32.2 ft/sec\textsuperscript{2}).

Step 3: Determine average velocity during first half of time period

The average velocity in the first half of the time period (from 0 to \( \frac{1}{2}T \)) can be calculated by averaging the initial velocity and the velocity at \( \frac{1}{2}T \): \( v_{avg1} = \frac{0 + g \left( \frac{1}{2} T \right)}{2} = \frac{0 + \frac{1}{2} g T}{2} = \frac{1}{4} g T \)

Step 4: Determine average velocity during second half of time period

The average velocity in the second half of the time period (from \( \frac{1}{2}T \) to T) can be calculated by averaging the velocity at \( \frac{1}{2}T \) and the velocity at T: \( v_{avg2} = \frac{g \left( \frac{1}{2} T \right) + g T}{2} = \frac{\frac{1}{2} g T + g T}{2} = \frac{\frac{3}{2} g T}{2} = \frac{3}{4} g T \)

Step 5: Compare the two average velocities

To show that the average velocity during the first half is one-third of the average velocity during the second half, divide \( v_{avg1} \) by \( v_{avg2} \): \[ \frac{v_{avg1}}{v_{avg2}} = \frac{\frac{1}{4} g T}{\frac{3}{4} g T} = \frac{1}{3} \] Thus, the average velocity during the first half is indeed one-third of the average velocity during the second half.

Key Concepts

average velocity

Average velocity is a way to describe how fast something is moving on average over a certain time period. It is calculated by taking the total displacement (or change in position) and dividing it by the total time taken. In the context of free fall motion, the average velocity over a time interval can be found by averaging the initial and final velocities during that interval. For instance, if a ball is dropped from rest (meaning its initial velocity is zero), the average velocity during the first half of its fall can be computed by looking at the velocity it reaches halfway through the fall.

kinematic equations

Kinematic equations are formulas that describe the motion of objects under the influence of constant acceleration. These equations help us calculate various aspects of motion, such as displacement, velocity, and time. For an object in free fall, the velocity at any time can be computed using the equation: \[ v = g t \] where \( v \) is the velocity, \( g \) is the acceleration due to gravity, and \( t \) is the time. In our exercise, we used this equation to find the velocity at different points in time and then used these velocities to calculate the average velocities.

acceleration due to gravity

Acceleration due to gravity, denoted by \( g \), is the rate at which objects accelerate towards the Earth when in free fall. The value of \( g \) is approximately 32.2 ft/sec\textsuperscript{2} and is constant near the Earth's surface. This means that for every second an object is in free fall, its velocity increases by 32.2 ft/sec. In the example problem, we see how the velocity of a falling ball changes over time due to this constant acceleration. Using the kinematic equation \( v = g t \), we can calculate how fast the ball is moving at any point in its fall.

initial velocity

Initial velocity is the speed at which an object starts its motion. In the context of the given problem, the ball is dropped from rest, meaning its initial velocity is zero. This initial condition is crucial because it simplifies the calculations and helps us better understand the concepts. With an initial velocity of zero, the average velocity calculations for the first and second halves of the fall are straightforward. Without this simplification, additional terms would be introduced, making the problem more complex. Starting from rest helps to focus on how gravity alone influences the falling object.