Question

The decreases in the potential energy of a ball of mass \(25 \mathrm{~kg}\) which falls from a height of \(40 \mathrm{~cm}\) is (A) \(968 \mathrm{~J}\) (B) \(100 \mathrm{~J}\) (C) \(1980 \mathrm{~J}\) (D) \(200 \mathrm{~J}\)

Short answer

The decrease in potential energy is calculated using the formula \(PE = m * g * h\). Given the mass (m) of the ball as \(25 kg\) and the height (h) as \(40 cm\), we first convert the height to meters (\(0.4 m\)). The initial potential energy is \(PE_{initial} = (25 kg) * (9.8 \frac{m}{s^2}) * (0.4 m) = 98 Nm\). Since the ball falls to the ground, the final potential energy is \(0 Nm\). The decrease in potential energy is \(PE_{initial} - PE_{final} = 98 Nm - 0 Nm = 98 Nm\). None of the given options match this value, so the correct answer should be approximately \(98 J\).

Step-by-step solution

Understand the problem and recall the formula for potential energy

We are given the mass (m) of the ball as 25 kg and the height (h) it falls from as 40 cm. We need to find the decrease in potential energy, which can be calculated using the formula for potential energy: Potential energy (PE) = m * g * h, where g represents the acceleration due to gravity (approximately 9.8 m/s²).

Convert height to meters

We are given the height in centimeters, so we need to convert it to meters before performing the calculation: \(h = 40 cm = 0.4 m\)

Calculate the initial potential energy

Now we can calculate the initial potential energy of the ball before it falls, using the formula PE = m * g * h with the provided values for mass and height, and a value of 9.8 m/s² for acceleration due to gravity: \(PE_{initial} = (25 kg) * (9.8 \frac{m}{s^2}) * (0.4 m) = 98 Nm\)

Calculate the final potential energy

Since the ball falls to the ground, its final height is 0 meters. Therefore, the final potential energy will be zero: \(PE_{final} = (25 kg) * (9.8 \frac{m}{s^2}) * (0 m) = 0 Nm\)

Determine the decrease in potential energy

Finally, to find the decrease in potential energy, we subtract the final potential energy from the initial potential energy: \(Decrease~in~potential~energy = PE_{initial} - PE_{final} = 98 Nm - 0 Nm = 98 Nm\) Comparing this value with the given options: (A) \(968 \mathrm{~J}\) (B) \(100 \mathrm{~J}\) (C) \(1980 \mathrm{~J}\) (D) \(200 \mathrm{~J}\) None of these options match the decrease in potential energy calculated. There may be an error in the given options, or the problem statement might be incorrect. Based on the calculation, the correct answer should be approximately 98 J.

Key concepts

Mechanics

Mechanics is a foundational branch of physics that focuses on the behavior of objects in motion and the forces that affect these objects. In the context of our exercise, the ball’s fall involves several key mechanical concepts that include motion, force, and energy.

A crucial aspect of mechanics is understanding how forces, such as gravitational force, impact the movement of an object. This involves analyzing the object's initiation of movement, its path, and eventual stopping point. In this exercise, as the ball drops from a height, it transitions from a state of rest to motion due to gravity acting upon it.

Understanding these foundational principles can help decipher real-world phenomena, like why things fall or how they gain speed. This lays the groundwork for exploring more complex topics in physics, including momentum and energy conservation.

Conservation of Energy

Conservation of energy is a vital principle in physics, stating that energy in a closed system remains constant. Energy changes forms but does not disappear. In our problem, it is central to understanding how potential energy becomes kinetic energy.

Initially, the ball has potential energy due to its elevated position. As it falls, this potential energy is converted to kinetic energy, causing the ball to gain speed. Upon reaching the ground, its potential energy effectively becomes zero.

To calculate the decrease in potential energy, we use the formula: \[ PE_{initial} = m \times g \times h,\] where \( m \) is mass, \( g \) is the gravitational acceleration (9.8 m/s²), and \( h \) is height. For the ball, the decrease in potential energy from a height of 0.4 m results in a calculation of approximately 98 J, demonstrating conservation by showing energy conversion from potential to kinetic form.

Gravitational Force

Gravitational force is a natural phenomenon wherein objects with mass attract each other, like the Earth pulling objects toward its center. It is this force that causes our ball to fall from the height.

Gravitational force acts as an unseen hand that accelerates the ball downwards. This force is quantified by the equation: \[ F = m \times g,\] where \( F \) is the gravitational force, \( m \) is the mass, and \( g \) is the acceleration due to gravity. In our exercise, \( g \) is standardized at approximately 9.8 m/s², affecting the ball's potential energy calculation.

An understanding of gravity is crucial for studying dynamics and energy transformations, aiding in predicting how objects will move under its influence. This isn't just limited to falling objects but extends to orbital mechanics and tides affected by celestial gravitational pull.