A toy plane has a mass of 2.5 kg and is 18 m above the ground. It is moving at [tex]$4.5 m / s$[/tex]. How much mechanical energy does the toy plane have?
A. 416 J
B. 441 J
C. 466 J
D. 491 J
Answer (2)
Calculate the potential energy using PE = m g h = 2.5 × 9.8 × 18 = 441 J.
Calculate the kinetic energy using K E = 2 1 m v 2 = 2 1 × 2.5 × ( 4.5 ) 2 = 25.3125 J.
Calculate the total mechanical energy by summing the potential and kinetic energies: ME = PE + K E = 441 + 25.3125 = 466.3125 J.
The total mechanical energy of the toy plane is approximately 466 J.
Explanation
Problem Analysis We are given a toy plane with a mass of 2.5 kg, a height of 18 m above the ground, and a speed of 4.5 m/s. We need to find the total mechanical energy of the toy plane. The mechanical energy is the sum of its potential energy (PE) and kinetic energy (KE).
Calculating Potential Energy First, let's calculate the potential energy (PE) of the toy plane. The formula for potential energy is PE = m g h , where m is the mass, g is the acceleration due to gravity (approximately 9.8 m / s 2 ), and h is the height.
Potential Energy Calculation Plugging in the values, we have: PE = ( 2.5 kg ) × ( 9.8 m/s 2 ) × ( 18 m ) PE = 441 J So, the potential energy of the toy plane is 441 J.
Calculating Kinetic Energy Next, let's calculate the kinetic energy (KE) of the toy plane. The formula for kinetic energy is K E = 2 1 m v 2 , where m is the mass and v is the speed.
Kinetic Energy Calculation Plugging in the values, we have: K E = 2 1 × ( 2.5 kg ) × ( 4.5 m/s ) 2 K E = 2 1 × 2.5 × 20.25 K E = 25.3125 J So, the kinetic energy of the toy plane is 25.3125 J.
Total Mechanical Energy Calculation Now, we can find the total mechanical energy (ME) by summing the potential and kinetic energies: ME = PE + K E ME = 441 J + 25.3125 J ME = 466.3125 J Therefore, the total mechanical energy of the toy plane is approximately 466.3125 J.
Final Answer The total mechanical energy of the toy plane is approximately 466.3125 J. Among the given options, 466 J is the closest value.
Examples
Understanding mechanical energy is crucial in various real-world applications. For instance, when designing roller coasters, engineers need to calculate the potential and kinetic energy at different points to ensure the coaster has enough energy to complete the track safely. Similarly, in sports, understanding the mechanical energy involved in a jump or throw helps athletes optimize their performance. In renewable energy, calculating the mechanical energy of wind or water is essential for designing efficient turbines.
The total mechanical energy of the toy plane, calculated by summing its potential and kinetic energy, is approximately 466 J. Therefore, the correct answer is 466 J. Among the options provided, this matches with the correct choice of 466 J.
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