Calculating Momentum Of A Running Horse A Physics Problem

In the realm of physics, understanding momentum is crucial for analyzing the motion of objects. Momentum, a fundamental concept, quantifies an object's mass in motion. It's the product of an object's mass and its velocity, expressed in kilogram-meters per second (kg m/s). This article delves into the concept of momentum and provides a step-by-step guide to calculating the momentum of a 76-kilogram horse running at a speed of 45 kilometers per hour. We will explore the formula for momentum, the importance of unit conversion, and the practical application of these calculations. This understanding is essential not just for students of physics but also for anyone interested in the mechanics of movement and the world around us. This exploration will not only clarify the concept of momentum but also demonstrate its practical application in real-world scenarios, such as analyzing the motion of animals or vehicles.

Defining Momentum and Its Significance

At its core, momentum is a measure of how difficult it is to stop a moving object. It depends both on the object's mass and its velocity. A heavier object moving at the same speed as a lighter one will have greater momentum, and similarly, an object moving faster will have more momentum than the same object moving slower. This relationship is mathematically expressed as:

p = mv

Where:

• p represents momentum (measured in kg m/s)
• m represents mass (measured in kilograms, kg)
• v represents velocity (measured in meters per second, m/s)

The significance of momentum extends beyond simple calculations. It is a conserved quantity in a closed system, meaning that the total momentum of a system remains constant if no external forces act on it. This principle is fundamental in understanding collisions, explosions, and other interactions between objects. For instance, in a collision between two cars, the total momentum of the two-car system before the collision is equal to the total momentum after the collision. This conservation law allows physicists and engineers to predict the outcome of such interactions and design systems that can withstand or mitigate the effects of impacts.

Problem Statement: Calculating the Momentum of a Running Horse

Our specific problem involves calculating the momentum of a 76-kilogram horse running at a speed of 45 kilometers per hour. This scenario provides a practical context for applying the momentum formula. To solve this problem, we need to first ensure that all units are consistent. The standard unit for velocity in the momentum calculation is meters per second (m/s), while the given speed is in kilometers per hour (km/h). Therefore, a unit conversion is necessary before we can apply the formula. This conversion is a critical step in many physics problems, as using inconsistent units will lead to incorrect results. The process of converting units involves multiplying the given value by a conversion factor, which is a ratio that expresses the equivalence between different units. In this case, we will convert kilometers per hour to meters per second by multiplying by the appropriate conversion factor.

Step 1: Unit Conversion Kilometers per Hour to Meters per Second

Before we can calculate the momentum, we need to convert the horse's speed from kilometers per hour (km/h) to meters per second (m/s). This conversion is crucial because the standard unit for velocity in physics calculations, particularly those involving momentum, is meters per second. Using kilometers per hour directly in the formula would lead to an incorrect result. The conversion factor we need is derived from the fact that 1 kilometer is equal to 1000 meters and 1 hour is equal to 3600 seconds. Therefore, to convert from km/h to m/s, we multiply by the ratio (1000 meters / 1 kilometer) / (3600 seconds / 1 hour), which simplifies to 1000/3600 or 5/18. Applying this conversion factor to the horse's speed:

• Speed in m/s = 45 km/h * (1000 m / 1 km) / (3600 s / 1 h)
• Speed in m/s = 45 * (5/18) m/s
• Speed in m/s = 12.5 m/s

This conversion is not just a mathematical necessity; it also provides a more intuitive sense of the horse's speed. 12. 5 meters per second is a tangible measure of how far the horse travels each second, making it easier to conceptualize the horse's motion. This step highlights the importance of paying attention to units in physics problems and ensuring that they are consistent before performing calculations.

Step 2: Applying the Momentum Formula

Now that we have the horse's speed in meters per second (12.5 m/s) and its mass (76 kg), we can calculate its momentum using the formula p = mv. This formula is the cornerstone of momentum calculations and directly relates the momentum of an object to its mass and velocity. By substituting the known values into the formula, we can determine the horse's momentum. This step is a direct application of the theoretical concept of momentum to a practical scenario. The accurate substitution of values and the correct application of the formula are critical for obtaining the correct answer. The formula itself is a concise representation of a fundamental physics principle, and its application in this context demonstrates the power of mathematical models in describing real-world phenomena.

• p = 76 kg * 12.5 m/s
• p = 950 kg m/s

Step 3: Expressing the Answer in Scientific Notation

The calculated momentum of the horse is 950 kg m/s. However, the answer choices are presented in scientific notation, which is a way of expressing numbers as a product of a number between 1 and 10 and a power of 10. Scientific notation is particularly useful for expressing very large or very small numbers concisely. To express 950 in scientific notation, we need to rewrite it in the form a × 10^b, where 1 ≤ |a| < 10 and b is an integer. In this case, we move the decimal point three places to the left, resulting in 9.5, and multiply by 10 raised to the power of 2. This conversion to scientific notation is a standard practice in scientific calculations and is essential for comparing our result with the given answer choices. It also demonstrates an understanding of numerical representation and the ability to express quantities in different formats.

• 950 kg m/s = 9.5 x 10^2 kg m/s

Comparing our calculated momentum, 9.5 x 10^2 kg m/s, with the given answer choices, we find that option D, 9.5 x 10^2 kilogram-meters/second, is the correct answer. This step is the culmination of the entire problem-solving process, where we match our calculated result with the available options. The ability to correctly identify the matching answer demonstrates not only an understanding of the physics concepts and calculations but also the ability to interpret and compare numerical results. This final step reinforces the importance of accuracy and attention to detail in scientific problem-solving. The process of elimination may also be used if other answers can be clearly ruled out based on magnitude or units.

• A. 2.5 x 10^2 kilogram-meters/second (Incorrect)
• B. 3.4 x 10^4 kilogram-meters/second (Incorrect)
• C. 3.4 x 10^3 kilogram-meters/second (Incorrect)
• D. 9.5 x 10^2 kilogram-meters/second (Correct)

In summary, we have successfully calculated the momentum of a 76-kilogram horse running at 45 kilometers per hour. The process involved converting the speed from kilometers per hour to meters per second, applying the momentum formula (p = mv), and expressing the result in scientific notation. The correct answer is 9.5 x 10^2 kilogram-meters/second. This exercise highlights the importance of understanding fundamental physics concepts, paying attention to units, and applying mathematical formulas accurately. Furthermore, it demonstrates the practical application of momentum calculations in real-world scenarios. The ability to solve such problems is not only valuable for students of physics but also for anyone interested in understanding the mechanics of motion and the world around us. This comprehensive approach to problem-solving, from understanding the underlying concepts to performing the calculations and interpreting the results, is a key skill in physics and other scientific disciplines. The understanding of momentum gained through this exercise can be applied to a wide range of situations, from analyzing the motion of vehicles to understanding collisions and impacts.