Moving objects, such as automobiles, have kinetic energy (KE), which is measured in joules. This energy can be calculated using the following formula:
[tex]$KE=\frac{1}{2} m v^2$[/tex]
A person drives an automobile with a mass of 450 kilograms at a velocity of 26 meters per second. The driver accelerates to a velocity of 30 meters per second.
The difference in the automobile's kinetic energy between the two velocities is $\square$ joules.
Answer (2)
Calculate the initial kinetic energy: K E 1 = 2 1 × 450 × 2 6 2 .
Calculate the final kinetic energy: K E 2 = 2 1 × 450 × 3 0 2 .
Calculate the difference in kinetic energy: Δ K E = K E 2 − K E 1 = 2 1 × 450 × ( 3 0 2 − 2 6 2 ) .
The difference in kinetic energy is 50400 joules.
Explanation
Understanding the Problem We are given the formula for kinetic energy, K E = 2 1 m v 2 , where m is the mass and v is the velocity. We need to find the difference in kinetic energy when the velocity changes from 26 m/s to 30 m/s, given the mass of the automobile is 450 kg.
Calculating Initial Kinetic Energy First, we calculate the initial kinetic energy K E 1 when the velocity is 26 m/s: K E 1 = 2 1 m v 1 2 = 2 1 × 450 × 2 6 2
Calculating Final Kinetic Energy Next, we calculate the final kinetic energy K E 2 when the velocity is 30 m/s: K E 2 = 2 1 m v 2 2 = 2 1 × 450 × 3 0 2
Calculating the Difference in Kinetic Energy Now, we find the difference in kinetic energy, Δ K E = K E 2 − K E 1 :
Δ K E = K E 2 − K E 1 = 2 1 × 450 × 3 0 2 − 2 1 × 450 × 2 6 2 = 2 1 × 450 × ( 3 0 2 − 2 6 2 ) Δ K E = 2 1 × 450 × ( 900 − 676 ) = 2 1 × 450 × 224 = 225 × 224 = 50400
Final Answer The difference in the automobile's kinetic energy between the two velocities is 50400 joules.
Examples
Kinetic energy is a fundamental concept in physics and engineering. For example, when designing vehicles, engineers need to calculate the kinetic energy involved in collisions to improve safety features. Understanding how changes in velocity affect kinetic energy helps in designing efficient braking systems and crumple zones that absorb energy during accidents, reducing the impact on passengers. This principle is also crucial in designing roller coasters, where changes in height and velocity convert potential energy into kinetic energy and vice versa, providing a thrilling ride experience.
The difference in the automobile's kinetic energy when its velocity changes from 26 m/s to 30 m/s is 51300 joules. This is calculated using the formula for kinetic energy by finding the energy at both speeds and determining the difference. The mass of the automobile is 450 kg, which factors into the calculations of kinetic energy.
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