One Graph to Rule Them All: Unlocking Newton’s Second Law Through Hands-On Experiment

I’m Ken Kuwako, the Science Trainer. Every day is an experiment.

“What is the single most important equation in high school physics?”

If someone asked you that question, how would you answer? Some might point to the famous equation associated with Einstein’s theory of relativity, while others might choose an equation representing the conservation of energy. There are plenty of worthy candidates. But if you asked many physicists to pick just one, chances are they would choose the famous equation of motion, F = ma.

Just three characters. Yet within this simple equation lies the motion of a thrown ball, the acceleration of a car, and even the moment a rocket launches into space. Knowing an equation and truly experiencing it are two very different things. In this article, I’ll introduce an experiment that allows students to discover this equation for themselves through hands-on investigation.

Purpose of the Experiment: Exploring the Relationship Between Mass, Acceleration, and Force

The goal of this experiment is to uncover the relationship between force (F), mass (m), and acceleration (a) through real experimental data. Ultimately, students will work toward deriving the famous equation $F=ma$ on their own. Instead of simply being handed a formula, they get the excitement of discovering it.

Preparation Tips: Setting Up for Success

This experiment uses a device called a constant-force apparatus. As the name suggests, it continuously pulls an object with nearly the same force whether the object is stationary or already moving. When students try pulling by hand, the force naturally fluctuates, making it difficult to obtain consistent data. A constant-force apparatus removes that variable and produces much more reliable results.

Most constant-force apparatuses come with two wires. For example, the “Constant Force Apparatus DJ-0461” provides approximately $0.49\text{N}$ (roughly the weight of 50 g) through the upper wire and approximately $0.98\text{N}$ (roughly the weight of 100 g) through the lower wire. If you measure the pulling force using a spring scale, you’ll find slight variations, but the device maintains a remarkably consistent force overall.

Constant Force Apparatus DJ-0461

Some teachers may think, “That sounds great, but it’s a bit expensive.” At around ¥10,000, it’s not exactly an impulse purchase. However, if your school can provide one for each lab group, the quality of the data improves dramatically. You can perform a similar experiment using a spring scale while trying to maintain a constant pull, but if you want accurate and reproducible results, a constant-force apparatus is well worth considering.

The “Magic” of Using Two Speed Sensors

Many science teachers are familiar with the speed-measuring device known as the Bee-Spi. Using a photoelectric sensor, it records the time an object passes through and calculates its speed.

Normally, it’s used to measure the speed of an object passing through a sensor. But if you place two Bee-Spi units in sequence, you can accurately determine an object’s acceleration.

The setup is surprisingly simple. Measure and fix the distance between the two sensors, then attach a chopstick to the front of the cart. As the chopstick passes through the first and second sensors, the cart’s initial and later speeds are recorded. Using those values and the equation below,

you can calculate the otherwise invisible quantity known as acceleration. Check out the video below to see the experiment in action.

The Science Recipe: Materials and Procedure

Let’s walk through the setup and procedure.

Materials

• Constant-force apparatus: DJ-0461 or equivalent (verify the force values of your own model)
• Dynamics cart
• Bee-Spi speed sensors: Two units (for example, BeeSpi V)
• 1-meter ruler: For measuring the sensor spacing
• Disposable wooden chopstick
• Modeling clay: To add mass (for example, three 500 g blocks)
• Electronic balance: For measuring clay mass accurately
• Newton spring scale: Capable of measuring up to about 2 N
• Calculator: For acceleration calculations
• Vinyl tape: For securing components

Experimental Procedure

1. Prepare the spring scale: Hold it horizontally and adjust it so the pointer reads $0\text{N}$.
2. Measure the force produced by the constant-force apparatus: Use the spring scale to determine the pulling force of each wire and record the values. These numbers will become important during analysis.
   • Upper wire: $()\text{N}$
   • Lower wire: $()\text{N}$
3. Prepare the cart: Attach a chopstick to the cart using tape. The chopstick will pass through the Bee-Spi sensors and also serve as the hook point for the constant-force apparatus (see the figure below).

1. Run the trials and collect data: Measure the cart’s speeds using the two Bee-Spi sensors and calculate acceleration under the following seven conditions. Use the constant-acceleration equation v2−v02​=2ax. Assume the cart itself has a mass of $1.0\text{kg}$.

• ① No experiment (control condition).
• ② No added mass, pulled by the upper wire only.
• ③ No added mass, pulled by the lower wire only.
• ④ No added mass, pulled by both wires simultaneously.
• ⑤ Add a 500 g mass and pull using the lower wire only.
• ⑥ Add a 1000 g mass and pull using the lower wire only.
• ⑦ Add a 1500 g mass and pull using the lower wire only.

Analysis and Discussion: Discovering the Law Through Graphs

Once the data has been collected, it’s time to visualize the relationships and see what patterns emerge.

• Create an a–F graph: Using data from conditions ①–④, plot acceleration (a) on the vertical axis and force (F) on the horizontal axis.
• Create an a–m graph: Using data from conditions ③, ⑤, ⑥, and ⑦, plot acceleration (a) versus mass (m).
• Create an a–1/m graph: Using the same data, plot acceleration (a) against the reciprocal of mass (1/m). This graph reveals something particularly important.

The results are often striking:

• The a–F graph shows a clear direct proportionality. Double the force, and the acceleration doubles.
• The a–m graph reveals an inverse relationship. The larger the mass, the harder it is to accelerate.
• The a–1/m graph produces an impressively straight proportional relationship.

Combining these findings leads to

F=kma​

where k is a proportionality constant. If we define a force of 1 as the amount required to accelerate a 1 kg object at 1 m/s², then k becomes 1, and we arrive at the celebrated equation:

F = ma

At this point, you can also explain that the unit of force, the Newton (N), is defined based on this relationship. Students gain a much deeper understanding of what “force” actually means rather than treating it as just another quantity in a formula.

Seeing Motion with Stroboscopic Photography

What’s the difference between giving an object a brief push and continuously applying a force?

The following stroboscopic images make the answer visible. Watch the video below.

The top sequence shows uniform motion, while the bottom sequence shows accelerated motion. Both objects are moving, but the pattern of motion changes dramatically depending on how force is applied. Being able to make that difference visible is one of the most satisfying parts of this activity.

Through this experiment, students do far more than memorize a formula. They investigate a fundamental law of nature, uncover patterns from real data, and experience the thrill of discovering one of physics’ most important ideas for themselves. Give it a try in your classroom—you may be surprised by how much it sparks students’ curiosity about physics.

Contact and Collaboration

Bring the wonder of science closer to everyday life! This site is packed with fun science experiments you can try at home, along with practical tips and explanations. Feel free to explore.

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Can Students Discover F = ma on Their Own? A Hands-On Physics Experiment That Reveals Newton’s Greatest Equation

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