Calculate the kinetic energy of a proton moving with a velocity of 2 imes 10^6 m/s. (Mass of proton = 1.67 \

Question

Calculate the kinetic energy of a proton moving with a velocity of 2 	imes 10^6 m/s. (Mass of proton = 1.67 	imes 10^{-27} kg).

Examoogle · Accepted Answer

The correct answer is Option A: $3.34 	imes 10^{-15} 	ext{ J}$. Kinetic energy is the energy possessed by an object due to its motion. The kinetic energy of the proton is calculated using its mass and the square of its velocity according to the classical mechanics formula.

Step-by-Step Solution & Concepts:
Kinetic energy is the energy possessed by an object due to its motion. The kinetic energy of the proton is calculated using its mass and the square of its velocity according to the classical mechanics formula.

Working Principle: According to the work-energy theorem, the kinetic energy ($K$) of an object of mass $m$ moving with velocity $v$ is given by $K = \frac{1}{2}mv^2$. Substituting the given values: $K = \frac{1}{2} 	imes (1.67 	imes 10^{-27} 	ext{ kg}) 	imes (2 	imes 10^6 	ext{ m/s})^2$. Solving this, we get $K = 0.5 	imes 1.67 	imes 10^{-27} 	imes 4 	imes 10^{12} = 3.34 	imes 10^{-15} 	ext{ J}$.

Key Formulas: $K = \frac{1}{2}mv^2$ — Formula for kinetic energy where $m$ is mass and $v$ is velocity.

Key Points: Kinetic energy is directly proportional to the mass of the object.; Kinetic energy is directly proportional to the square of the velocity ($v^2$), meaning a small increase in speed leads to a large increase in energy.; The SI unit of kinetic energy is the Joule ($J$), where $1 	ext{ J} = 1 	ext{ kg} \cdot 	ext{m}^2/	ext{s}^2$.

Advantages: Useful for calculating energy transitions in subatomic particles at non-relativistic speeds.

Disadvantages: This classical formula is invalid for particles moving near the speed of light; relativistic kinetic energy must be used instead.

Applications: Particle accelerators for studying high-energy physics.; Understanding chemical bonding and ionization processes.

Additional Info: The proton mass is approximately $1.6726 	imes 10^{-27} 	ext{ kg}$.. Option B ($6.68 	imes 10^{-15} 	ext{ J}$) results from forgetting the $\frac{1}{2}$ factor in the formula.. Option C ($1.67 	imes 10^{-15} 	ext{ J}$) results from neglecting the $v^2$ term correctly or calculation errors.. Option D ($3.34 	imes 10^{-14} 	ext{ J}$) results from a decimal shift error in the power of 10.

Answer: A is correct — The calculated kinetic energy using $K = \frac{1}{2}mv^2$ is $3.34 	imes 10^{-15} 	ext{ J}$.

Exam Tip: Always ensure your units are in SI (kg, m/s) before plugging them into the formula to avoid errors in powers of 10.

Examoogle · Answer

Option B: $6.68 	imes 10^{-15} 	ext{ J}$

Examoogle · Answer

Option C: $1.67 	imes 10^{-15} 	ext{ J}$

Examoogle · Answer

Option D: $3.34 	imes 10^{-14} 	ext{ J}$

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