An 8.00 kg box sits on a level floor. You give the box a...

An $8.00 \mathrm{~kg}$ box sits on a level floor. You give the box a sharp push and find that it travels $8.22 \mathrm{~m}$ in $2.8 \mathrm{~s}$ before coming to rest again. (a) You measure that with a different push the box traveled $4.20 \mathrm{~m}$ in $2.0 \mathrm{~s}$. Do you think the box has a constant acceleration as it slows down? Explain your reasoning. (b) You add books to the box to increase its mass. Repeating the experiment, you give the box a push and measure how long it takes the box to come to rest and how far the box travels. The results, including the initial experiment with no added mass, are given in the table: In each case, did your push give the box the same initial speed? What is the ratio between the greatest initial speed and the smallest initial speed for these four cases? (c) Is the average horizontal force $f$ exerted on the box by the floor the same in each case? Graph the magnitude of force $f$ versus the total mass $m$ of the box plus its contents, and use your graph to determine an equation for $f$ as a function of $m$

An $8.00 \mathrm{~kg}$ box sits on a level floor. You give the box a sharp push and find that it travels $8.22 \mathrm{~m}$ in $2.8 \mathrm{~s}$ before coming to rest again. (a) You measure that with a different push the box traveled $4.20 \mathrm{~m}$ in $2.0 \mathrm{~s}$. Do you think the box has a constant acceleration as it slows down? Explain your reasoning. (b) You add books to the box to increase its mass. Repeating the experiment, you give the box a push and measure how long it takes the box to come to rest and how far the box travels. The results, including the initial experiment with no added mass, are given in the table:
In each case, did your push give the box the same initial speed? What is the ratio between the greatest initial speed and the smallest initial speed for these four cases? (c) Is the average horizontal force $f$ exerted on the box by the floor the same in each case? Graph the magnitude of force $f$ versus the total mass $m$ of the box plus its contents, and use your graph to determine an equation for $f$ as a function of $m$

Key Concepts

Uniform Acceleration

This concept refers to motion in which the acceleration remains constant over time. Under uniform acceleration, the standard kinematic equations can be applied to relate displacement, initial velocity, time, and acceleration. Analyzing whether the box decelerates uniformly involves checking if the measured distances and times are consistent with a constant rate of speed reduction, which is a common assumption in introductory mechanics problems.

Kinematic Equations

Kinematic equations describe the relationships between displacement, initial velocity, acceleration, and time for objects undergoing constant acceleration. They provide a mathematical framework to predict one variable when others are known. In the context of the problem, these equations are used to analyze the motion of the box and to determine if the experimental measurements are consistent with the assumption of constant acceleration.

Newton’s Second Law

Newton’s Second Law states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration (F = ma). This principle is crucial when evaluating how changes in mass affect acceleration and, consequently, the forces experienced by the box. In the experiment, it helps in analyzing whether the horizontal force from the floor remains constant as mass is added, by relating the observed deceleration to the applied force through varying total masses.

Impulse and Momentum

Impulse and momentum are related through the impulse-momentum theorem, which states that the impulse applied to an object equals the change in its momentum. When the box is given a push, it gains an initial velocity that is proportional to the impulse delivered, independent of the subsequent deceleration details. Assessing whether the initial speeds are the same under different conditions involves understanding that, if the impulse is identical, the resulting momentum—and thus the initial speed—should be consistent, assuming little energy loss during the push.

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