Calculating The Impact Speed Of A Falling Ball Disregarding Air Resistance

This article delves into the physics behind calculating the impact speed of an object falling from a certain height, specifically addressing the scenario of a ball dropped from 12 feet. We will disregard air resistance to simplify the calculations and focus on the fundamental principles of kinematics. This problem is a classic example of applying constant acceleration equations, which are essential for understanding motion under the influence of gravity. By converting the height from feet to meters and utilizing the standard acceleration due to gravity, we can accurately determine the ball's velocity just before it strikes the ground.

Understanding the Physics Principles

At the heart of this calculation lies the principle of uniformly accelerated motion. This principle dictates that an object falling freely under the influence of gravity experiences a constant acceleration, denoted as g, which is approximately 9.8 m/s² on Earth. The absence of air resistance allows us to treat gravity as the sole force acting on the ball, simplifying our analysis. The kinematic equations, derived from the principles of calculus and Newtonian mechanics, provide the tools to relate displacement, initial velocity, final velocity, acceleration, and time for objects moving with constant acceleration. The specific equation we'll use, v² = u² + 2as, elegantly connects the final velocity (v) to the initial velocity (u), acceleration (a), and displacement (s), allowing us to directly compute the impact speed without explicitly calculating the time of fall.

When addressing motion problems under constant acceleration, a systematic approach is crucial for achieving accurate solutions. First, clearly define the initial conditions. In this scenario, the ball starts from rest, implying an initial velocity (u) of 0 m/s. The displacement (s) is given as 12 feet, which we'll convert to meters using the provided conversion factor. The acceleration (a) is the acceleration due to gravity (g), 9.8 m/s². Next, select the appropriate kinematic equation. Considering that we know the initial velocity, displacement, and acceleration, and we aim to find the final velocity, the equation v² = u² + 2as is perfectly suited. By substituting the known values into this equation and performing the necessary algebraic manipulations, we can isolate the final velocity (v). Finally, calculate the result and ensure that the units are consistent and physically meaningful. A negative sign for velocity would indicate a downward direction, which aligns with the problem's context. By adhering to this structured approach, we can confidently navigate a wide range of motion problems and deepen our understanding of kinematics.

Step-by-Step Calculation

To determine the speed of the ball just before impact, we'll follow a step-by-step approach, ensuring clarity and accuracy in our calculations. First, convert the height from feet to meters. We are given that 1 foot equals 0.30 meters. Therefore, 12 feet is equivalent to 12 ft * 0.30 m/ft = 3.6 meters. This conversion is crucial because the acceleration due to gravity is provided in meters per second squared (m/s²), and we need consistent units throughout our calculations. Using the metric system ensures that our final velocity will also be in meters per second (m/s), a standard unit for measuring speed.

Next, we apply the appropriate kinematic equation. As discussed earlier, the equation v² = u² + 2as is ideal for this scenario. Here, v represents the final velocity (what we want to find), u is the initial velocity (0 m/s since the ball is dropped), a is the acceleration due to gravity (9.8 m/s²), and s is the displacement (3.6 meters). Substituting these values into the equation gives us v² = 0² + 2 * 9.8 m/s² * 3.6 m. Simplifying this expression, we get v² = 70.56 m²/s². To find the final velocity v, we take the square root of both sides: v = √(70.56 m²/s²) ≈ 8.4 m/s. The positive square root is taken since we are interested in the magnitude of the velocity (speed) just before impact. This result represents the ball's speed as it is about to hit the ground, neglecting the effects of air resistance.

Detailed Solution and Explanation

Let's break down the solution to this problem step-by-step to ensure a comprehensive understanding. The problem states that a ball is dropped from a height of 12 feet, and we need to find its speed just before it hits the ground, disregarding air resistance. We are given the conversion factor 1 ft = 0.30 m and the acceleration due to gravity g = 9.8 m/s². This is a classic kinematics problem that can be solved using the equations of motion under constant acceleration.

The first crucial step is to convert the height from feet to meters. We multiply the height in feet by the conversion factor: 12 ft * 0.30 m/ft = 3.6 meters. This conversion ensures that all our units are consistent, as the acceleration due to gravity is given in meters per second squared. Using consistent units is vital for accurate calculations in physics problems. If we were to mix units, such as using feet for height and meters per second squared for acceleration, our final result would be incorrect.

Next, we identify the relevant kinematic equation. In this case, we can use the equation v² = u² + 2as, where v is the final velocity, u is the initial velocity, a is the acceleration, and s is the displacement. This equation is particularly useful because it allows us to find the final velocity without explicitly calculating the time it takes for the ball to fall. We know that the ball is dropped, meaning its initial velocity u is 0 m/s. The acceleration a is the acceleration due to gravity, which is 9.8 m/s². The displacement s is the distance the ball falls, which we have calculated to be 3.6 meters.

Substituting these values into the equation, we get v² = 0² + 2 * 9.8 m/s² * 3.6 m. Simplifying this expression, we have v² = 0 + 70.56 m²/s², which means v² = 70.56 m²/s². To find the final velocity v, we take the square root of both sides: v = √(70.56 m²/s²) ≈ 8.4 m/s. The result is approximately 8.4 meters per second. This represents the speed of the ball just before it hits the ground, considering the acceleration due to gravity and the height from which it was dropped.

Conclusion: Final Velocity Calculation

In conclusion, by applying the principles of kinematics and carefully converting units, we have determined that the speed of the ball just before it hits the ground is approximately 8.4 m/s. This calculation underscores the importance of understanding constant acceleration and the utility of kinematic equations in solving real-world physics problems. The ability to accurately calculate the impact speed of a falling object has numerous practical applications, ranging from engineering design to safety analysis. Understanding the concepts discussed here provides a solid foundation for further exploration of mechanics and dynamics.

Therefore, the correct answer is B. 8.4 m/s.