Question

If the mass of a truck is doubled- for example, when it is loaded - by what factor does the kinetic energy of the truck increase? By what factor does the kinetic energy change if the mass is one-tenth of the original mass?

Short answer

The kinetic energy increases by a factor of 2 when the mass is doubled, and it changes to one-tenth of the original value when the mass is one-tenth.

Step-by-step solution

Understanding the Kinetic Energy Formula

The kinetic energy (KE) of an object is given by the formula KE = \( \frac{1}{2}mv^2 \), where m represents the mass of the object, and v represents its velocity. Since only the mass of the truck is changing and velocity remains constant, we can focus on how changes in mass affect the kinetic energy.

Calculating the Factor of Increase for Doubling the Mass

If the mass of the truck is doubled, the new mass is \(2m\). The energy will then be \(KE_{new} = \frac{1}{2}(2m)v^2 = 2\left(\frac{1}{2}mv^2\right) = 2 \times KE\). Hence, the kinetic energy of the truck increases by a factor of 2.

Calculating the Kinetic Energy for One-Tenth the Mass

If the mass of the truck is one-tenth of the original, the new mass is \(\frac{1}{10}m\). The new kinetic energy is \(KE_{new} = \frac{1}{2}\left(\frac{1}{10}m\right)v^2 = \frac{1}{10}\left(\frac{1}{2}mv^2\right) = \frac{1}{10} \times KE\). Therefore, the kinetic energy of the truck becomes one-tenth of the original value.

Key concepts

Mass and Kinetic Energy Relationship

Understanding the relationship between mass and kinetic energy is crucial in fields like mechanics and engineering. Kinetic energy, the energy an object possesses due to its motion, is directly proportional to its mass. This implies that if you increase an object's mass, its kinetic energy will also increase, assuming its velocity remains constant.

Let's illustrate this with a practical example: imagine a truck moving along a highway. If the truck is initially carrying no cargo and we then load it with goods, effectively doubling its mass, we would expect the truck's kinetic energy to also double, if the speed remains unchanged. Similarly, reducing the truck's mass to one-tenth would mean its kinetic energy also reduces to one-tenth. This proportional change occurs because kinetic energy is calculated based on the mass of the object in motion. Such an understanding helps in making predictions about energy requirements and safety measures in transport and other motion-related industries.

Kinetic Energy Formula

To calculate the kinetic energy of an object in motion, we use the formula: \[KE = \frac{1}{2}mv^2\]Here, \( KE \) stands for kinetic energy, \( m \) is the mass of the object, and \( v \) represents the velocity at which the object is moving.

Breaking Down the Formula

The factor \( \frac{1}{2} \) is a constant that comes from the work-energy principle in physics. Mass \( m \) is a measure of how much matter is present in the object, and it is reported in kilograms (kg). Velocity \( v \) is the speed of the object in a specified direction, measured in meters per second (m/s) in the SI unit system.

This formula shows that kinetic energy is dependent not only on how fast an object is moving but also on its mass. For example, a slow-moving truck can have the same kinetic energy as a small, fast-moving sports car, depending on the values of mass and velocity.

Factors Affecting Kinetic Energy

Kinetic energy is influenced by two primary factors: mass and velocity, as we've seen in the kinetic energy formula. However, the dependence on these factors differs significantly. The velocity component in the equation is squared, which means that kinetic energy will increase exponentially with an increase in velocity. This explains why even a slight increase in speed can lead to a substantial rise in kinetic energy, making high-speed impacts so dangerous.

Implications on Energy and Safety

An object with a large mass has substantial kinetic energy at a given speed, but if that speed increases even slightly, the kinetic energy will ramp up dramatically. This relationship is why safety measures in vehicles, such as airbags and crumple zones, are so important; they help manage the kinetic energy during impacts. In various applications, understanding the factors that affect kinetic energy is essential for designing efficient systems that are both safe and energy-conscious.