Skateboarder Velocity Calculation Physics Problem Solved

In the realm of physics, understanding the relationship between momentum, mass, and velocity is crucial. This article delves into a practical problem involving a skateboarder to illustrate these concepts. We will explore how to calculate the velocity of a skateboarder given their momentum and mass, applying fundamental physics principles to arrive at the solution. We'll break down the relevant formulas, discuss the steps involved in the calculation, and interpret the result in the context of real-world scenarios. This exercise will not only solidify your understanding of these core physics concepts but also demonstrate their application in everyday situations.

Momentum in physics is a measure of the mass in motion: It’s fundamentally a vector quantity, meaning it has both magnitude and direction. The momentum of an object is defined as the product of its mass and velocity. Mathematically, this is expressed as:

p = m v

Where:

• p represents momentum (typically measured in kg-m/s)
• m represents mass (typically measured in kg)
• v represents velocity (typically measured in m/s)

The formula reveals that momentum is directly proportional to both mass and velocity. This means that an object with a larger mass or a higher velocity will have a greater momentum. Consider a scenario involving two objects, a car and a bicycle, both moving at the same velocity. The car, possessing a significantly larger mass, will have a considerably higher momentum than the bicycle. Conversely, if we consider two objects with the same mass, the object moving at a higher velocity will have a greater momentum. Think about a baseball thrown by a pitcher versus the same baseball gently tossed; the thrown baseball has a much higher velocity and thus a greater momentum.

In various real-world scenarios, understanding momentum is crucial. In sports, for example, athletes manipulate momentum to achieve desired outcomes. A football player running at high speed has a substantial momentum, making them harder to stop. Similarly, in car crashes, the momentum of the vehicles involved plays a critical role in determining the severity of the impact. A high-speed collision involves a large change in momentum over a short period, resulting in significant forces.

The principle of conservation of momentum is another vital concept. It states that the total momentum of an isolated system (one where no external forces are acting) remains constant. This principle is particularly useful in analyzing collisions. For instance, when two billiard balls collide, the total momentum before the collision is equal to the total momentum after the collision, assuming no external forces like friction are significant. This allows us to predict the motion of the balls after the impact.

In summary, momentum is a fundamental concept in physics that describes the quantity of motion an object possesses. It depends on both the mass and velocity of the object, and its understanding is crucial in analyzing various physical phenomena, from everyday scenarios to complex scientific problems. The equation p = m v is a cornerstone in mechanics, providing a simple yet powerful tool for understanding the motion of objects.

In this specific problem, we are given the following information:

• Momentum (p) of the skateboarder = 100 kg-m/s
• Mass (m) of the skateboarder = 15 kg

Our goal is to find the velocity (v) of the skateboarder. To do this, we will use the momentum formula we discussed earlier:

p = m v

To find the velocity, we need to rearrange the formula to solve for v. This involves dividing both sides of the equation by the mass (m).

Starting with the momentum formula:

p = m v

To isolate v, we divide both sides of the equation by m:

v = p / m

Now we substitute the given values into the equation:

v = (100 kg-m/s) / (15 kg)

Performing the division:

v = 6.666... m/s

The problem asks us to round the answer to the nearest hundredth, so we round 6.666... m/s to 6.67 m/s.

Let's walk through the solution step-by-step to ensure clarity and understanding. First, we begin with the fundamental equation for momentum:

1. The Momentum Equation: The foundation of our calculation is the formula p = m v, where p represents momentum, m represents mass, and v represents velocity. This equation succinctly captures the relationship between these three essential physical quantities. Momentum, as we've discussed, is a measure of an object's mass in motion, and it directly correlates with both the object's mass and its velocity. A heavier object moving at the same speed as a lighter one will have more momentum, and an object moving faster will have more momentum than the same object moving slower.

2. Rearranging the Formula: Our objective is to find the velocity, v, given the momentum and mass. To do this, we need to algebraically manipulate the momentum equation. We start with p = m v. To isolate v, we perform the inverse operation of multiplication, which is division. We divide both sides of the equation by m:

   • p / m = (m v) / m
   • This simplifies to v = p / m. This rearranged formula is now perfectly suited for our calculation, as it expresses velocity in terms of momentum and mass.

3. Substituting the Given Values: The problem provides us with the momentum (p) of the skateboarder, which is 100 kg-m/s, and the mass (m) of the skateboarder, which is 15 kg. We now substitute these values into our rearranged formula:

   • v = 100 kg-m/s / 15 kg
   • This step is crucial as it translates the abstract physics equation into a concrete calculation with specific numerical values.

4. Performing the Calculation: Now we perform the division to find the numerical value of the velocity:

   • v = 100 / 15
   • v = 6.666... m/s
   • This division yields a repeating decimal, which we will address in the next step by rounding.

5. Rounding to the Nearest Hundredth: The problem specifies that we should round our answer to the nearest hundredth. The hundredth place is the second digit after the decimal point. In our result, 6.666..., the digit in the hundredth place is 6, and the digit to its right (in the thousandth place) is also 6. Since this digit is 5 or greater, we round up the hundredth place.

   • Therefore, 6.666... m/s rounded to the nearest hundredth is 6.67 m/s.
   • Rounding is an essential step in many scientific and engineering calculations, as it allows us to present results with an appropriate level of precision, avoiding the implication of greater accuracy than is warranted.

Therefore, the velocity of the skateboarder is approximately 6.67 m/s. This corresponds to option D in the given choices.

The result makes intuitive sense when we consider the values of momentum and mass. A momentum of 100 kg-m/s for a 15 kg object suggests a moderate speed. If the mass were much larger, the velocity would be smaller for the same momentum, and vice versa. This reinforces our understanding of the inverse relationship between mass and velocity when momentum is constant.

It’s important to note the units in this calculation. Momentum is given in kg-m/s, mass in kg, and the resulting velocity is in m/s. Maintaining consistent units is crucial in physics calculations to ensure the accuracy and physical relevance of the results.

In this article, we successfully calculated the velocity of a skateboarder using the principles of momentum. We began by defining momentum and its relationship to mass and velocity. We then set up the problem, rearranged the momentum formula to solve for velocity, and substituted the given values. After performing the calculation and rounding to the nearest hundredth, we arrived at the solution: 6.67 m/s. This exercise demonstrates the practical application of fundamental physics concepts in solving real-world problems. Understanding these principles is essential for anyone studying physics or related fields. The ability to apply formulas, manipulate equations, and interpret results is a key skill in scientific problem-solving.

The final answer is (D) 6.67 m / s