1.  Introduction

Acceleration is a fundamental concept in physics, which refers to the rate of change of an object's velocity over time. It is very important in classical mechanics and contains Newton's second law that F=ma. Acceleration is a very basic and important physical quantity in physics. It connects force and mass and motion. Acceleration is an important role and has wide application value in engineering, aerospace and other fields.

Despite its foundational importance, recent studies have highlighted persistent challenges in students' comprehension of Newton's Second Law. Rosiqoh and Suhendi conducted a study using the Rasch model to assess physics education students' understanding of Newton's Second Law and found that 97% of participants exhibited low mastery of the concept, indicating significant gaps in conceptual understanding [1]. Similarly, Addido et al. explored the use of LEGO EV3 robots in teaching Newton's Second Law and reported that integrating robotics into instruction enhanced students' conceptual understanding and increased their interest in STEM fields [2].

This study aims to investigate the relationship between acceleration, force, and mass through a theoretical and review-based approach. The significance of this study lies in its potential to reinforce foundational physics concepts through theoretical analysis, especially in educational settings where access to laboratory resources may be limited. By providing a clear and structured examination of Newton's Second Law, this work contributes to the broader effort of enhancing physics education and addressing common misconceptions related to force and motion.

2.  Theoretical background

The definition of acceleration is the change of the velocity with the time.

The formula is

$$ \mathrm{a}=\frac{\mathrm{\delta }\mathrm{v}}{\mathrm{\delta }\mathrm{t}}$$(1)

where  $$ \mathrm{a}$$  is acceleration, the  $$ \mathrm{\delta }\mathrm{v}$$  is the change of the variation quantity of the velocity, and the  $$ \mathrm{\delta }\mathrm{t}$$  means the change of the variation quantity of the time.

The Newton’s second law indicate that the acceleration is inversely relationship to the mass and it directly relationship to the force. The formula is F=ma.

But if the object is not in the horizontal plane, it moving on an inclined plane is also affected by the gravitational component and the friction force. Assuming the inclination of the inclined plane is theta the, acceleration of the object can be expressed as:

$$ \mathrm{a}=\mathrm{g}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{\theta }-\mathrm{\mu }\mathrm{g}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{\theta }$$(2)

Where g is acceleration, and  $$ \mathrm{\mu }$$  is frictional coefficient.

Recent advancements have explored the emergence of Newtonian mechanics from statistical principles. Yuan and Sun derived Newton's Second Law from the statistical distribution of particles in a dilute gas, demonstrating that the law naturally arises when the system reaches equilibrium [3]. They interpreted the external force as a measure of distribution inhomogeneity, providing a novel perspective on the law's foundation.

In the realm of field theory, Schwebel proposed a generalization of Newton's Second Law applicable to parameterized submanifolds of higher dimensions [4]. By introducing the geodesic kkk-vector field, they extended the concept of inertial motion and established a connection between generalized Newtonian dynamics and Hamilton's canonical equations in field theory.

From an educational standpoint, Wang et al. utilized Atwood's machine to verify Newton's Second Law in a classroom setting [5]. Their approach simplified the experimental setup and reduced measurement errors, making it more accessible for students to understand the practical implications of the law.​

These studies highlight the multifaceted nature of Newton's Second Law, encompassing theoretical derivations, generalizations, and educational applications. Understanding experiment for verifying Newton's second law.lationship between force, mass, and acceleration.

3.  Experimental method

In classical mechanics, the relationship among force, mass and acceleration is usually proved by standard physical experiments. This article summarizes a simple experiment for verifying Newton's second law.

A widely used apparatus for this purpose includes a low-friction cart, a pulley, and a set of hanging weights. The car is placed on a horizontal track to minimize friction. One end of a lightweight string is attached to the cart, and the other end passes over a pulley to suspend a known mass. When the mass is released, it falls under the influence of gravity, applying a constant force to the cart via the string.

To conduct the experiment, first place the car on a track and connect one end of it to a thin rope. The rope should pass over a pulley and be suspended by a heavy object at the other end. Ensure that the pulley rotates freely to minimize friction. Begin by measuring the mass of the car, and prepare a set of weights with different masses to apply varying forces—while ignoring the friction of the pulley and the mass of the rope. Select a specific weight, record its mass, and release the car, allowing it to accelerate due to the gravitational pull on the weight. To measure the car’s acceleration, use either a photoelectric gate or a dot timer. If using a photoelectric gate, measure the time it takes for the car to pass between two gates and calculate the acceleration based on the known distance. If using a dot timer, analyze the spacing between the dots on the paper tape to determine acceleration. Repeat the measurement several times and calculate the average acceleration. Then, change the force by using different weights, repeating the procedure each time. It is important to ensure that the total mass of the car, including any added weights, remains constant throughout all trials.

Newton's second law precisely describes the relationship among force, mass, and acceleration under macroscopic and low-speed conditions. The high consistency between experimental results and theoretical predictions has proved its authenticity and universality. It has been demonstrated that when other factors remain unchanged, the resultant force is directly proportional to acceleration, and mass is inversely proportional to acceleration. While this law holds under many everyday conditions, more complex physical experiments are needed to test its applicability in high-speed or microscopic fields.

To investigate these relationships, motion sensors such as photoelectric gates or tick timers are typically used to measure acceleration. A photoelectric gate places two light beams along the track at a known distance; the system records the time it takes for a car or trolley to pass through both gates, allowing acceleration to be calculated using motion equations. Alternatively, a tick timer records motion by generating dots on a moving tape attached to the trolley. The spacing of these dots reflects the change in velocity over time, from which acceleration can be derived.

By changing the weights to alter the force, a comparison can be made under the condition that the mass remains constant. Specifically, different masses can be hung on the pulley to vary the gravitational force applied. In these tests, it is important to ensure that the total mass of the trolley system remains unchanged, so the effect of the force alone can be studied. Conversely, to examine the effect of mass on acceleration, heavy objects can be added to the trolley while keeping the hanging mass constant.

Overall, repeated experiments using motion sensors under controlled conditions provide strong evidence supporting Newton’s second law. Whether varying force or mass, keeping other variables fixed allows for clear observation of their respective effects on acceleration. This consistency between experiment and theory further underscores the law’s reliability under classical mechanics.

4.  Results and discussion

According to the equation F=ma, acceleration is directly proportional to the net force acting on an object and inversely proportional to its mass.

4.1.  Theoretical data

To examine the relationship between acceleration and applied force, consider a system with a total constant mass of 0.5 kg. As the applied force increases from 1 N to 5 N in increments of 1 N, the expected acceleration values would be shown in Table 1.

This trend illustrates a linear relationship between force and acceleration when the mass is held constant. A plot of this data would produce a straight line passing through the origin, supporting the theoretical prediction.

4.2.  Discussion

While the results demonstrate a clear linear relationship between force and acceleration when mass is held constant—consistent with Newton’s second law—some potential sources of error must be considered when applying this model to real-world experiments.

First, friction in the pulley system may introduce systematic error. In an ideal case, the pulley is assumed to be frictionless, but in practice, internal resistance within the pulley bearings can reduce the net force acting on the system, leading to an observed acceleration that is lower than the theoretical value.

Second, air resistance, although often negligible at low speeds, may still exert a small opposing force on the moving trolley. This resistance becomes more noticeable as acceleration (and hence velocity) increases, subtly affecting the measurements—especially at higher force values.

Third, the mass of the connecting rope is typically ignored in theoretical calculations. However, if the rope has non-negligible mass, it adds to the total inertia of the system unevenly as it shifts position during motion, which could alter the effective mass and, consequently, the measured acceleration.

Additionally, timing device inaccuracies may also contribute to deviations from theoretical predictions. For example, photoelectric gates and tick timers rely on precise triggering mechanisms. Any delay in sensor activation or variation in tape movement speed (in the case of tick timers) may slightly distort the calculated acceleration.

Moreover, friction between the trolley and the track—even if minimized—could further reduce acceleration, particularly if the track is not perfectly level or clean.

Despite these potential sources of error, the overall linear trend observed in the data and the close match between expected and calculated values affirm the fundamental validity of Newton's second law. Although real-world results may deviate slightly due to unavoidable experimental imperfections, the principle that acceleration is directly proportional to force and inversely proportional to mass remains robust and reliable under controlled conditions.

5.  Conclusion

This study has explored the core relationship described by Newton’s second law through theoretical analysis and hypothetical data. By maintaining a constant mass and varying the applied force, a clear linear relationship between force and acceleration was demonstrated, aligning with classical mechanics predictions. The results support the foundational idea that acceleration is directly proportional to net force and inversely proportional to mass.

However, this investigation has certain limitations. Since the analysis is based solely on idealized conditions and hypothetical data, it lacks real experimental measurements. As a result, potential errors such as friction, air resistance, and the mass of connecting components like pulleys and ropes were not quantitatively assessed. This restricts the ability to evaluate the accuracy and practical applicability of the findings.

Future work could address these limitations by conducting real-world experiments to gather empirical data. Incorporating actual measurements would allow for error analysis and a more comprehensive understanding of the law’s application. In addition, exploring more complex scenarios, such as motion on inclined planes, variable mass systems, or non-uniform acceleration, could deepen the analysis and reflect more realistic physics situations. Despite its simplifications, this study affirms the reliability of Newton’s second law as a fundamental principle in classical mechanics.