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Chapter 17: Problem 13

(II) What is the speed of a proton whose \(\small\mathrm{KE}\) is 4.2 keV?

Short Answer

Expert verified
The speed of the proton is approximately \( 8.97 \times 10^5 \text{ m/s} \).

Step by step solution

01

Understand Units of Energy

The kinetic energy (KE) provided is in kiloelectronvolts (keV), which needs to be converted to joules (J) because the standard unit for energy in physics is the joule. Recall that \( 1 \, \text{eV} = 1.602 \times 10^{-19} \, \text{J} \). Thus, \( 4.2 \, \text{keV} = 4.2 \times 10^3 \, \text{eV} \) and must be converted to joules.
02

Convert Kinetic Energy to Joules

Convert the kinetic energy from keV to J by multiplying:\[ \text{KE} = 4.2 \times 10^3 \times 1.602 \times 10^{-19} \, \text{J} = 6.7276 \times 10^{-16} \, \text{J}. \]
03

Use Kinetic Energy Formula

The formula for the kinetic energy of a particle is \( \text{KE} = \frac{1}{2} m v^2 \). We need to solve this for the velocity \( v \). Rearrange the formula to find:\[ v = \sqrt{\frac{2 \times \text{KE}}{m}}. \]
04

Insert Proton Mass

The mass of a proton is known to be \( m = 1.6726 \times 10^{-27} \, \text{kg} \). Substitute this value and the kinetic energy into the velocity equation:\[ v = \sqrt{\frac{2 \times 6.7276 \times 10^{-16} \, \text{J}}{1.6726 \times 10^{-27} \, \text{kg}}}. \]
05

Calculate the Velocity

Calculate the square root to find the velocity:\[ v = \sqrt{\frac{1.34552 \times 10^{-15}}{1.6726 \times 10^{-27}}} = \sqrt{8.0416 \times 10^{11}} \, \text{m/s}. \]Finally, compute the square root to obtain:\[ v \approx 8.97 \times 10^5 \, \text{m/s}. \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Kinetic Energy
Kinetic energy is a vital concept in physics. It describes the energy a particle possesses due to its motion. The formula to calculate kinetic energy is given by: \( \text{KE} = \frac{1}{2} m v^2 \). Here, \( m \) represents the mass and \( v \) denotes the velocity of the particle. The kinetic energy tells us how much work is needed to accelerate an object from rest to its current speed. In our problem, we are given the kinetic energy in kiloelectronvolts (keV), which we convert into joules for more precise calculations. Understanding kinetic energy helps us predict the behavior of moving particles, such as protons in this example.
Unit Conversion
Unit conversion is essential when dealing with different measurement systems. In physics, consistency in units is crucial to ensure correct results. For energy, the standard unit is the joule (J), but sometimes, values are given in kiloelectronvolts (keV). We convert keV to joules to avoid errors. One kiloelectronvolt equals \( 1.602 \times 10^{-19} \) joules. Therefore, to convert \(4.2 \text{ keV} \) to joules, we multiply the energy by \( 1.602 \times 10^{-19} \), resulting in \( 6.7276 \times 10^{-16} \) J. By using consistent units, we ensure calculations, like finding the speed of a proton, are accurate and meaningful.
Proton Mass
The mass of a proton is a fundamental constant in physics. Protons are subatomic particles found in the nucleus of atoms, and they have a mass of approximately \( 1.6726 \times 10^{-27} \) kg. Knowing the proton's mass is essential when calculating its speed from the kinetic energy equation. In our calculations, substituting the proton mass into the equation for velocity \( v = \sqrt{\frac{2 \times \text{KE}}{m}} \) allows us to determine how fast the proton is traveling. This mass value is critical for solving various physics problems involving particles and plays a significant role in atomic and particle physics.

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Most popular questions from this chapter

Paper has a dielectric constant \(K =\) 3.7and a dielectric strength of 15 \(\times\) 10\(^6\)V/m. Suppose that a typical sheet of paper has a thickness of 0.11 mm. You make a "homemade" capacitor by placing a sheet of 21 \(\times\) 14cm paper between two aluminum foil sheets (Fig. 17-48) of the same size. (a) What is the capacitance \(C\) of your device? (\(b\)) About how much charge could you store on your capacitor before it would break down?

A parallel-plate capacitor with plate area \(A =\) 2.0 m\(^2\) and plate separation \(d =\) 3.0 mm is connected to a 35-V battery (Fig. 17-51a). (\(a\)) Determine the charge on the capacitor, the electric field, the capacitance, and the energy stored in the capacitor. (\(b\)) With the capacitor still connected to the battery, a slab of plastic with dielectric strength \(K =\) 3.2 is placed between the plates of the capacitor, so that the gap is completely filled with the dielectric (Fig. 17-51b).What are the new values of charge, electric field, capacitance, and the energy stored in the capacitor?

A huge 4.0-F capacitor has enough stored energy to heat 2.8 kg of water from 21\(^\circ\)C to 95\(^\circ\)C. What is the potential difference across the plates?

(II) Figure 17-44 is a photograph of a computer screen shot by a camera set at an exposure time of \(\frac{1}{4}\)s. During the exposure the cursor arrow was moved around by the mouse, and we see it 15 times. (\(a\)) Explain why we see the cursor 15 times. (\(b\)) What is the refresh rate of the screen?

II) There is an electric field near the Earth's surface whose magnitude is about 150 V/m. How much energy is stored per cubic meter in this field?
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