Ball Drop: Speed At Impact From 12 Feet

Let's dive into a classic physics problem: determining the speed of a ball just before it hits the ground after being dropped from a height of 12 feet. We'll disregard air resistance to keep things simple and focus on the fundamental principles of kinematics. To solve this, we'll use the given conversions and the acceleration due to gravity.

Understanding the Problem

When an object is dropped, it accelerates downwards due to gravity. The acceleration due to gravity, denoted as g, is approximately 9.8 m/s². This means that for every second the ball falls, its speed increases by 9.8 meters per second. We need to find the final velocity (v) of the ball just before it impacts the ground, given the initial height (h) and the acceleration due to gravity (g).

Converting Units

First, we need to convert the height from feet to meters. We are given that 1 foot is equal to 0.30 meters. Therefore, a height of 12 feet is:

$$h = 12 \text{ feet} \times 0.30 \frac{\text{meters}}{\text{foot}} = 3.6 \text{ meters}$$

Applying Kinematic Equations

We can use one of the basic kinematic equations to solve this problem. The equation that relates final velocity (v), initial velocity (u), acceleration (a), and displacement (s) is:

$$v^2 = u^2 + 2as$$

In our case:

• v is the final velocity (what we want to find).
• u is the initial velocity. Since the ball is dropped, the initial velocity is 0 m/s.
• a is the acceleration, which is the acceleration due to gravity, g = 9.8 m/s².
• s is the displacement, which is the height h = 3.6 meters.

Plugging these values into the equation, we get:

$$v^2 = 0^2 + 2 \times 9.8 \times 3.6$$

$$v^2 = 0 + 70.56$$

$$v^2 = 70.56$$

Now, we take the square root of both sides to find v:

$$v = \sqrt{70.56}$$

$$v \approx 8.4 \text{ m/s}$$

So, the speed of the ball just before it hits the ground is approximately 8.4 m/s.

Detailed Explanation and Elaboration

Initial Conditions and Assumptions

In this problem, we made a key assumption: we disregarded air resistance. In reality, air resistance would play a significant role, especially for objects with a large surface area or low mass. Air resistance is a force that opposes the motion of an object through the air, and it increases with the speed of the object. If we were to include air resistance, the ball would reach a terminal velocity, which is the maximum speed it can achieve while falling. At terminal velocity, the force of air resistance equals the force of gravity, and the net force on the object is zero, resulting in no further acceleration.

Understanding Kinematic Equations

Kinematic equations are a set of equations that describe the motion of objects with constant acceleration. These equations are derived from the definitions of displacement, velocity, and acceleration. The equation we used, $v^2 = u^2 + 2as$, is particularly useful when we don't know the time it takes for the object to travel from its initial position to its final position.

Step-by-Step Calculation

1. Convert Units: Convert the height from feet to meters using the given conversion factor.
2. Identify Known Variables: Identify the initial velocity, acceleration, and displacement.
3. Choose the Appropriate Kinematic Equation: Select the kinematic equation that relates the known variables to the unknown variable.
4. Plug in Values: Substitute the known values into the equation.
5. Solve for the Unknown Variable: Solve the equation for the unknown variable.

The Role of Gravity

Gravity is the force that pulls objects towards each other. On Earth, the acceleration due to gravity is approximately 9.8 m/s². This value can vary slightly depending on location, but for most calculations, 9.8 m/s² is a good approximation. The force of gravity is what causes the ball to accelerate downwards when it is dropped.

Implications of Ignoring Air Resistance

By ignoring air resistance, we simplified the problem and were able to use a basic kinematic equation to find the final velocity. However, it's important to remember that in real-world scenarios, air resistance can have a significant impact on the motion of objects. For example, a feather dropped from the same height would take much longer to reach the ground than a ball because the air resistance acting on the feather is much greater relative to its weight.

Alternative Methods

Another way to solve this problem would be to first find the time it takes for the ball to fall using the equation:

$$s = ut + \frac{1}{2}at^2$$

Where:

• s is the displacement (3.6 meters).
• u is the initial velocity (0 m/s).
• a is the acceleration (9.8 m/s²).
• t is the time.

Plugging in the values, we get:

$$3.6 = 0 \times t + \frac{1}{2} \times 9.8 \times t^2$$

$$3.6 = 4.9t^2$$

$$t^2 = \frac{3.6}{4.9}$$

$$t = \sqrt{\frac{3.6}{4.9}} \approx 0.857 \text{ seconds}$$

Then, we can use the equation:

$$v = u + at$$

Where:

• v is the final velocity.
• u is the initial velocity (0 m/s).
• a is the acceleration (9.8 m/s²).
• t is the time (0.857 seconds).

Plugging in the values, we get:

$$v = 0 + 9.8 \times 0.857$$

$$v \approx 8.4 \text{ m/s}$$

As you can see, we get the same result using this method.

Conclusion

In summary, by using the principles of kinematics and ignoring air resistance, we found that the speed of a ball dropped from 12 feet just before it hits the ground is approximately 8.4 m/s. This problem illustrates the basic concepts of motion under constant acceleration and highlights the importance of understanding the assumptions and limitations of our models. Understanding these concepts allows us to predict and analyze the motion of objects in a variety of scenarios.

Learn more about kinematics on Khan Academy. This resource provides further explanations and examples to enhance your understanding of motion in physics.