Question

An automobile has a mass of \(1200 \mathrm{~kg}\). What is its kinetic energy, in \(\mathrm{kJ}\), relative to the road when traveling at a velocity of \(50 \mathrm{~km} / \mathrm{h}\) ? If the vehicle accelerates to \(100 \mathrm{~km} / \mathrm{h}\), what is the change in kinetic energy, in \(\mathrm{kJ}\) ?

Short answer

The change in kinetic energy is approximately 347.13 kJ.

Step-by-step solution

- Convert velocities to meters per second

First, convert the velocities from \text{km/h} to \text{m/s}. Use the conversion factor: \(1 \text{ km/h} = \frac{1}{3.6} \text{ m/s}\).For 50 km/h:\[50 \text{ km/h} = 50 \times \frac{1}{3.6} = \frac{50}{3.6} \approx 13.89 \text{ m/s}\]For 100 km/h:\[100 \text{ km/h} = 100 \times \frac{1}{3.6} = \frac{100}{3.6} \approx 27.78 \text{ m/s}\]

- Calculate the initial kinetic energy

Calculate the kinetic energy at the initial velocity (\text{v} = 13.89 \text{ m/s}) using the formula for kinetic energy:\(KE = \frac{1}{2} m v^2\).\[KE_1 = \frac{1}{2} \times 1200 \text{ kg} \times (13.89 \text{ m/s})^2\]\[KE_1 \approx \frac{1}{2} \times 1200 \times 193.05 \approx 115830 \text{ J}\]Since 1 kJ = 1000 J, \[KE_1 \approx 115.83 \text{ kJ}\]

- Calculate the final kinetic energy

Calculate the kinetic energy at the final velocity (\text{v} = 27.78 \text{ m/s}) using the same formula:\(KE = \frac{1}{2} m v^2\).\[KE_2 = \frac{1}{2} \times 1200 \text{ kg} \times (27.78 \text{ m/s})^2\]\[KE_2 \approx \frac{1}{2} \times 1200 \times 771.60 \approx 462960 \text{ J}\]Since 1 kJ = 1000 J, \[KE_2 \approx 462.96 \text{ kJ}\]

- Calculate the change in kinetic energy

Find the change in kinetic energy (\text{ΔKE}) by subtracting the initial kinetic energy from the final kinetic energy:\[ΔKE = KE_2 - KE_1\]\[ΔKE = 462.96 \text{ kJ} - 115.83 \text{ kJ} \approx 347.13 \text{ kJ}\]

Key concepts

kinetic energy

Kinetic energy is the energy an object possesses due to its motion. It is an essential concept in physics, tying into everything from everyday movement to complex mechanical systems. The formula for kinetic energy (KE) is: \(KE = \frac{1}{2}mv^2\), where \(m\) is the mass of the object and \(v\) is its velocity. The higher the object's mass and velocity, the greater its kinetic energy. In practical terms, this means a speeding car has significantly more kinetic energy than a slower one or a lighter object moving at the same speed. Understanding this helps us predict how much work is needed to alter the object's movement, such as bringing a car to a stop.

unit conversion

Unit conversion is critical for making accurate calculations, particularly in physics and engineering. In our exercise, we needed to convert velocities from kilometers per hour (km/h) to meters per second (m/s) because the standard SI unit for speed is \(m/s\). The conversion factor is simple: \(1 \text{ km/h} = \frac{1}{3.6} \text{ m/s}\). Therefore, to convert:

• 50 km/h to m/s: \(50 \times \frac{1}{3.6} \) ≈ \(13.89 \text{ m/s}\)
• 100 km/h to m/s: \(100 \times \frac{1}{3.6} \) ≈ \(27.78 \text{ m/s}\)

This step ensures our calculations are within the correct framework, avoiding errors due to unit inconsistency.

velocity transformation

Velocity transformation is about expressing velocity in suitable units to meet specific needs. In this exercise, converting velocities from \(km/h\) to \(m/s\) was necessary to use the kinetic energy formula correctly. This translation not only helps standardize data but also simplifies complex equations in physics. Changing units correctly can also assist in comparing different physical quantities and making sense of experimental data. For instance, while km/h might be more intuitive for travel speed, m/s fits within the broader metric system used for scientific calculations, ensuring precision across different contexts.

mass-velocity relationship

The mass-velocity relationship is pivotal in understanding kinetic energy. Kinetic energy depends on both the mass (m) of the object and its velocity (v). From the formula: \(KE = \frac{1}{2}mv^2\), we see that kinetic energy increases linearly with mass but quadratically with velocity.

• Doubling the mass of an object doubles its kinetic energy.
• Doubling the velocity, however, increases the kinetic energy by four times.

This shows why speed is such a critical factor in energy dynamics. Small changes in velocity lead to significant changes in kinetic energy, making high-speed scenarios particularly energy-intensive.