Ball Rolling Downhill: Calculate Final Speed

by Alex Johnson 45 views

Let's explore how to determine the final speed of a ball rolling down a hill. We'll use a classic physics problem involving a 10 kg ball released from the top of a hill to illustrate the concepts and calculations involved. We'll approximate the acceleration due to gravity, denoted as 'g', as 10 m/s² for simplicity and round our final answer to the nearest tenth. This problem allows us to delve into the principles of energy conservation and how potential energy transforms into kinetic energy as the ball descends.

Understanding the Problem

Before diving into calculations, let's break down the problem and identify the key elements:

  • Mass of the ball (m): 10 kg
  • Acceleration due to gravity (g): Approximately 10 m/s²
  • Initial state: Ball at the top of the hill (potential energy is maximum, kinetic energy is zero)
  • Final state: Ball at the base of the hill (potential energy is minimum, kinetic energy is maximum)

We need to determine the final speed (v) of the ball when it reaches the base of the hill. Crucially, the problem does not provide the height of the hill. Therefore, we'll solve the problem in a general way, expressing the final velocity in terms of the height (h) of the hill. This makes our solution versatile – you can plug in any height value to find the corresponding final speed.

The Power of Energy Conservation

The cornerstone of solving this problem is the principle of energy conservation. In a closed system without non-conservative forces (like friction or air resistance), the total mechanical energy remains constant. Mechanical energy is the sum of potential energy (related to an object's position) and kinetic energy (related to an object's motion).

In our scenario, as the ball rolls down the hill, its potential energy (PE) is converted into kinetic energy (KE). At the top of the hill, the ball possesses maximum potential energy and zero kinetic energy (since it's initially at rest). Conversely, at the base of the hill, the ball has minimum potential energy (we can consider it zero at the base) and maximum kinetic energy. The energy conservation principle tells us:

Initial Potential Energy + Initial Kinetic Energy = Final Potential Energy + Final Kinetic Energy

Potential Energy

Potential energy (PE) is the energy an object possesses due to its position in a gravitational field. It is calculated as:

PE = mgh

where:

  • m = mass of the object (in kg)
  • g = acceleration due to gravity (in m/s²)
  • h = height of the object above a reference point (in meters)

In our case, the initial potential energy of the ball at the top of the hill is mgh, where h is the height of the hill.

Kinetic Energy

Kinetic energy (KE) is the energy an object possesses due to its motion. It is calculated as:

KE = (1/2)mv²

where:

  • m = mass of the object (in kg)
  • v = velocity of the object (in m/s)

At the top of the hill, the initial kinetic energy is zero because the ball is at rest. At the base of the hill, the final kinetic energy is (1/2)mv², where v is the final velocity we want to find.

Setting Up the Equation

Now, let's express the energy conservation principle mathematically for our problem:

mgh + 0 = 0 + (1/2)mv²

Here:

  • mgh represents the initial potential energy (at the top of the hill).
  • 0 represents the initial kinetic energy (at the top of the hill, ball is at rest).
  • 0 represents the final potential energy (at the base of the hill, we consider potential energy to be zero at the base).
  • (1/2)mv² represents the final kinetic energy (at the base of the hill).

Solving for the Final Velocity (v)

We can simplify the equation and solve for v:

mgh = (1/2)mv²

Notice that the mass m appears on both sides of the equation, so we can cancel it out:

gh = (1/2)v²

Now, multiply both sides by 2:

2gh = v²

Finally, take the square root of both sides to solve for v:

v = √(2gh)

This equation tells us that the final velocity of the ball at the base of the hill depends only on the acceleration due to gravity (g) and the height of the hill (h). It's independent of the mass of the ball!

Applying the Values

We know that g is approximately 10 m/s². Plugging this into the equation, we get:

v = √(2 * 10 * h)

v = √(20h)

This is our general solution. To get a numerical answer, we NEED the height of the hill, h. Since the problem doesn't provide h, we'll express the answer in terms of h.

Example:

Let's assume the height of the hill is 5 meters (h = 5 m). Then:

v = √(20 * 5)

v = √100

v = 10 m/s

So, if the hill is 5 meters high, the ball's final speed at the bottom would be 10 m/s.

Important: The problem asks us to round the answer to the nearest tenth. In our example, the answer is already a whole number, so no rounding is needed. However, if we had a height that resulted in a final velocity like 9.87 m/s, we would round it to 9.9 m/s.

The Importance of Height

The height of the hill is crucial for determining the final speed of the ball. Without knowing the height, we can only provide a general formula: v = √(20h). To get a specific numerical answer, you must know the value of h.

Considering Real-World Factors

It's important to acknowledge that this solution is based on ideal conditions. In the real world, factors like friction and air resistance would play a role. Friction between the ball and the surface of the hill would convert some of the mechanical energy into heat, reducing the ball's final kinetic energy and thus its final speed. Air resistance would also impede the ball's motion, further decreasing its final speed. Our simplified model neglects these factors to make the problem solvable with basic physics principles.

In Summary

To calculate the final speed of a 10 kg ball rolling down a hill, we use the principle of energy conservation. The potential energy at the top of the hill is converted into kinetic energy at the base. The final velocity is given by the formula v = √(2gh), where g is the acceleration due to gravity and h is the height of the hill. Remember to account for real-world factors like friction and air resistance in more complex scenarios. Since the height was not provided in the original problem, the final speed is expressed as a function of height: v = √(20h). If the height h were known, you could simply plug it into this formula and calculate the final velocity, rounding the answer to the nearest tenth as requested.

Understanding energy conservation and its applications helps us solve a wide array of physics problems, from simple scenarios like this one to more complex systems.

For further reading on energy conservation, check out this resource from Khan Academy.