Question

For these problems, we use Eq. \(1018, s=v_{0} t+a t^{2} / 2\) which says that, if assume the initial velocity is zero, the distance \(s\) fallen by a body (from rest) varies directly as the square of the elapsed time \(t\) If a body falls 738 units in \(3.00 \mathrm{s}\), how far will it fall in \(6.00 \mathrm{s} ?\)

Short answer

The body will fall 2952 units in 6.00 seconds.

Step-by-step solution

Identify the Known Values

The problem provides the distance fallen, which is 738 units when the time is 3.00 s. Since the body starts from rest, the initial velocity (\( v_0 \) is 0. We are tasked with finding the distance (\( s \) fallen in 6.00 s.

Apply the Equation for Distance Fallen

Use Eq. 1018, which is the formula for distance fallen from rest: \( s = v_0 t + a t^{2} / 2 \) Here, since \( v_0 = 0 \) (the initial velocity), the equation simplifies to \( s = a t^{2} / 2 \) where \( a \) is the acceleration due to gravity.

Calculate the Acceleration

First, solve for acceleration (\( a \) using the known distance and time for the first fall: \( 738 = a (3.00)^2 / 2 \). Multiply both sides by 2 to get \( 1476 = 9.00a \). Now, divide both sides by 9.00 to find \( a \) : \( a = 1476 / 9.00 = 164 \) units/sec^2.

Determine the Distance Fallen in 6.00 s

Now use the acceleration found in Step 3 to calculate the distance fallen in 6.00 s: \( s = 164 (6.00)^2 / 2 \). Simplify and calculate: \( s = 82 (36.00) \). Finally, \( s = 2952 \) units.

Key concepts

Uniformly Accelerated Motion

Understanding uniformly accelerated motion is foundational in physics, as it pertains to objects that experience constant acceleration. In our everyday experience, gravity is the most common source of constant acceleration.

For objects in freefall near the Earth's surface, this acceleration is approximately 9.81 meters per second squared, but the exercise abstracts this value. When an object is dropped and allowed to fall freely, it accelerates uniformly under the influence of gravity, assuming air resistance is negligible.

The body in our example problem exhibits uniformly accelerated motion, starting from rest. The distance it covers over time is not linear but rather it follows a quadratic relationship. This means that if the time were to double, the distance fallen would not merely double, but rather quadruple, considering the equation for distance is proportional to the square of time elapsed.

Kinematics Equations

Kinematics equations are the tools that allow us to predict the position, velocity, and acceleration of an object in motion under the umbrella of classical mechanics.

In this particular problem, we apply a simplified kinematic equation, given by Eq. 1018: \( s = v_0 t + \frac{1}{2}at^2 \), which describes the distance \( s \) covered by a body with initial velocity \( v_0 \), under uniform acceleration \( a \), after a certain time \( t \). When the initial velocity is zero, this equation reduces to \( s = \frac{1}{2}at^2 \), highlighting the direct proportionality between distance and the square of time.

This key equation is derived from the integrals of motion and encapsulates a core principle of kinematics, making it widely used for solving physics problems involving linear motion with uniform acceleration.

Physics Problem Solving

Physics problem solving is a step-by-step approach that allows us to analyze a situation and find a solution to the posed questions using physical laws and formulas.

To efficiently solve problems like the one about distance fallen, it's essential to:

• Identify and list known variables and what needs to be found.
• Carefully choose the appropriate physical formula that relates these variables.
• Carefully manipulate the formula to isolate and solve for the unknowns.
• Ensure the consistency of units throughout the calculations.
• Perform sanity checks on the results to see if they make physical sense.

In our example, proper utilization of the kinematics equation and algebra was the key to finding the correct solution. It demonstrates how an organized and methodical approach lends clarity to the task at hand and guides us to the right answer.