Question

Think about the answers to these questions next time you are driving a car: a) What is the kinetic energy of a \(1500 .-\mathrm{kg}\) car moving at \(15.0 \mathrm{~m} / \mathrm{s} ?\) b) If the car changed its speed to \(30.0 \mathrm{~m} / \mathrm{s}\), how would the value of its kinetic energy change?

Short answer

Answer: When the car's speed is doubled from 15.0 m/s to 30.0 m/s, its kinetic energy also doubles.

Step-by-step solution

Identify the given information

In this exercise, we are given: Mass of the car, m = 1500 kg Velocity of the car in situation a, v1 = 15.0 m/s Velocity of the car in situation b, v2 = 30.0 m/s

Calculate the initial kinetic energy (situation a)

Using the kinetic energy formula mentioned before, we can calculate the initial kinetic energy of the car (situation a) as follows: \(KE_1 = 0.5 * m * v_1^2\) \(KE_1 = 0.5 * (1500 kg) * (15.0 m/s)^2\) \(KE_1 = 0.5 * (1500 kg) * (225 m^2/s^2)\) \(KE_1 = 337500 J\) (Joules) So, the initial kinetic energy of the car is 337500 Joules.

Calculate the final kinetic energy (situation b)

Similarly, we can calculate the final kinetic energy of the car (situation b) using the same formula: \(KE_2 = 0.5 * m * v_2^2\) \(KE_2 = 0.5 * (1500 kg) * (30.0 m/s)^2\) \(KE_2 = 0.5 * (1500 kg) * (900 m^2/s^2)\) \(KE_2 = 675000 J\) (Joules) So, the final kinetic energy of the car is 675000 Joules.

Compare the kinetic energy values

The initial kinetic energy of the car (situation a) is 337500 Joules, while the final kinetic energy of the car (situation b) is 675000 Joules. To find how the value of kinetic energy has changed, we can find the ratio of the final kinetic energy to the initial kinetic energy: \(\frac{KE_2}{KE_1} = \frac{675000 J}{337500 J} = 2\) So, when the car's speed is doubled from 15.0 m/s to 30.0 m/s, its kinetic energy value has doubled.

Key concepts

Kinetic Energy Formula

Kinetic energy is a form of energy that an object possesses due to its motion. It's a key principle in classical mechanics, which states that moving objects can do work on other objects. To calculate the kinetic energy (\( KE \text{)}), we utilize the kinetic energy formula: \[ KE = \frac{1}{2} \times m \times v^2 \] wherein 'm' signifies the mass of the object (measured in kilograms) and 'v' represents the velocity (measured in meters per second).
By applying this formula, we can observe how kinetic energy depends squarely on the mass and the square of the velocity. As a practical example, consider a car in motion. If we take into account a car with a mass of 1500 kg traveling at 15.0 m/s, the kinetic energy of the car can be computed as follows: \[ KE_1 = \frac{1}{2} \times 1500 \text{kg} \times (15.0 \text{m/s})^2 = 337,500 \text{J} \] J, here, stands for Joules, which is the unit of energy. This straightforward formula is immensely helpful in predicting how much work an object in motion can potentially exert.

Work-Energy Principle

The work-energy principle is a fundamental concept in physics. It asserts that the work done on an object is equivalent to the change in its kinetic energy. 'Work' in a physical sense involves applying a force over a distance. When we apply a force to move a car, for instance, we're doing work on the car.
In mathematical terms, if an object's kinetic energy changes from \( KE_1 \) to \( KE_2 \) as a result of work \( W \) done on it, the principle can be expressed as: \[ W = KE_2 - KE_1 \] Adopting our previous example, when the car speeds up from 15.0 m/s to 30.0 m/s, its kinetic energy increases. This change in kinetic energy is attributed to the work done by the forces acting on the car (like the engine). The greater the speed, the more work is required to reach that velocity.

Conservation of Energy

The conservation of energy principle is another cornerstone of physics, which holds that within a closed system, energy can neither be created nor destroyed; it can only be transformed from one form to another. Kinetic energy can therefore be converted into other forms of energy, such as potential energy, heat, or light, but the total energy remains constant.
An example that illustrates this principle is a roller coaster at its peak about to descend; it has maximum potential energy and minimum kinetic energy. As it descends, potential energy is converted into kinetic energy. At the bottom, the roller coaster will have maximum kinetic energy and minimum potential energy. If we apply the conservation of energy principle to our car scenario from our exercise, any change in the car's speed (and therefore kinetic energy) must result from the conversion of another form of energy, typically stored chemical energy from fuel, into kinetic energy. The principle ensures that the total energy before and after the change in speed remains equal, even though the kinetic energy value itself has altered.