Question

A \(10 \mathrm{~kg}\) bowling ball falls from a height of \(5 \mathrm{~m}\). Using the convenient \(g \approx 10 \mathrm{~m} / \mathrm{s}^{2}\), how much gravitational potential energy does it have? Just before it hits the ground, all of this potential energy has gone into kinetic energy. \(^{34}\) What is the speed of the bowling ball when it reaches the ground, based on kinetic energy?

Short answer

Question: Calculate the gravitational potential energy of a 10 kg bowling ball falling from a height of 5 meters, and determine its speed when it reaches the ground. Answer: The gravitational potential energy of the bowling ball is 500 Joules, and its speed when it reaches the ground is 10 m/s.

Step-by-step solution

Calculating the Gravitational Potential Energy

The potential energy (PE) of an object can be calculated using the formula: PE = mgh. Here, 'm' portrays the mass of the bowling ball which is 10 kg, 'g' is approximately 10 m/s² which is the acceleration due to gravity, and 'h' is the height from which it falls, which in this case, is 5 m. The formula plugged in with these values becomes: PE = 10 kg * 10 m/s² * 5 m

Solving for the Gravitational Potential Energy

Now, by multiplying the values given in step 1, the result for the potential energy can be derived: PE = 10 kg * 10 m/s² * 5 m = 500 Joules

Setting up the Equation for Kinetic Energy

As per the law of conservation of energy, the potential energy of the ball just before it hits the ground equals to its kinetic energy. Thus, setting up the equation of kinetic energy as: 1/2 * m * v² = PE 1/2 * 10 kg * v² = 500 Joules Solving this for 'v' to find out the speed of the ball.

Solving for the Speed of the Bowling Ball

Since the aim is to find out 'v', the equation becomes: v² = 2 * PE / m Substitute the given values into this equation: v² = 2 * 500 J / 10 kg Solving this will yield the square of the speed of the ball. To extract the speed, take the square root of the result: v = sqrt(100)

Final Calculation of the Speed of the Bowling Ball

The square root of 100 is 10. Therefore, the speed of the bowling ball when it reaches the ground will be 10 m/s.

Key concepts

Kinetic Energy

Kinetic energy represents the energy an object possesses due to its motion. Every moving object, from the smallest particle to the largest spaceship, has kinetic energy. If you picture a bowling ball rolling down an alley, the energy it uses to knock over the pins is kinetic energy. Mathematically, kinetic energy (\( KE \)) is given by the equation \( KE = \frac{1}{2}mv^2 \), where \( m \) is the mass of the object and \( v \) is its velocity.

In the example of the falling bowling ball, as it drops, gravity does work on the bowling ball, accelerating it and thus increasing its kinetic energy. When the bowling ball starts from rest, all of its kinetic energy is initially in the form of gravitational potential energy. As it descends without air resistance, it converts this potential energy entirely into kinetic energy, until just before impact, its kinetic energy is equal to the original potential energy it had at the height of 5 meters.

Conservation of Energy

The principle of the conservation of energy is a cornerstone of physics and asserts that energy in a closed system remains constant—it cannot be created or destroyed, only transformed from one form to another. This immutable principle underlies a multitude of phenomena in the physical world.

When applying this concept to our bowling ball scenario, the energy we start with (gravitational potential energy) must be equal to the energy we end with (kinetic energy), if we neglect losses such as friction or air resistance. The initial gravitational potential energy that the ball possesses due to its position is converted into kinetic energy as the ball falls. As the ball gains speed, its kinetic energy increases while its gravitational potential energy decreases, but the total energy remains the same. The exercise demonstrates that the ball’s kinetic energy just before hitting the ground is 500 Joules, the same amount as its initially calculated potential energy.

Physics Education

Physics education aims to impart a deep understanding of the natural laws that govern our universe. It covers a vast range of topics, from mechanics and thermodynamics to quantum physics and relativity. One of the primary goals in teaching physics is to enable students to apply these fundamental principles, like the conservation of energy, to solve quantitative problems and to develop a conceptual grasp of physical processes.

To ensure that students can easily grasp these concepts, educators often use real-world examples, such as a falling bowling ball, to bridge the gap between abstract principles and tangible experiences. Interactive problem-solving, such as working through textbook exercises step-by-step, reinforces learning. In the solution provided, we observe a practical application of theoretical physics, empowering students to appreciate the predictive power of physical laws and to nurture problem-solving skills that are invaluable both in and out of the classroom.