Question 1

When one compares the relativistic kinetic energy to the classically calculated kinetic energy for a moving object,at which of the speeds listed below does the classical calculation give a greater value than the relativistic calculation?
A)0.2 c
B)0.5 c
C)0.707 c
D)None of the above values will give such a result.

Accepted Answer

The classical calculation of kinetic energy always gives a lesser value than the relativistic calculation when the speed of the object is a significant fraction of the speed of light (c). This is because relativistic effects, which include time dilation and length contraction, become significant at high speeds, causing the relativistic kinetic energy to be higher than the classical kinetic energy. Therefore, at none of the speeds listed will the classical calculation give a greater value than the relativistic calculation.

Question 2

A lump of uranium has a mass of 2.0 kg,and begins at rest.Half of the lump's mass is going to be totally converted into kinetic energy of the other half.After this is done,how fast is the remaining half going?
A)0.60 c
B)0.80 c
C)0.87 c
D)1.0 c

Accepted Answer

According to the principle of mass-energy equivalence ($E=mc^2$), the mass lost during the conversion process will be converted into energy. Using conservation of energy, we can set the initial potential energy equal to the final kinetic energy:

$Potential\ energy = \frac{1}{2}m_{final}v^2 + mc^2$

Half of the mass will be converted totally into kinetic energy of the other half, so $m_{final} = m_{initial}/2 = 1$ kg.

Substituting in the given values, we have:

$(2\ kg)(c^2) = \frac{1}{2}(1\ kg)v^2 + (2\ kg)(c^2)$

Simplifying, we get:

$\frac{1}{2}(2\ kg)(c^2) = \frac{1}{2}(1\ kg)v^2$

$v^2 = 2c^2$

$v = \sqrt{2}c \approx 0.87c$

Therefore, the remaining half is going at a speed of 0.87 times the speed of light, which is approximately 87% of the speed of light. The correct answer is (C).

Question 3

In an x-ray tube,high-speed electrons are slammed into a lead target,giving off x-rays.If the electrons are accelerated from rest through a potential difference of 50 000 volts,what speed do they have when they strike the target? (q_e = 1.6 x 10^-19 C,m_e = 9.11 x 10^-31 kg,and c = 3.00 x 10⁸ m/s) A)0.17 c B)0.41 c C)0.83 c D)0.91 c

Accepted Answer

The kinetic energy gained by the electron is equal to the electric potential energy, which is qV. Using the equation KE = 0.5 * m * v^2 = q * V, we can solve for v. Substituting the given values, v = sqrt((2 * 1.6 x 10^-19 C * 50,000 V) / (9.11 x 10^-31 kg)) ≈ 0.41c.

Question 4

If the mass of a proton is 1.67 x 10^-27 kg,what is the rest energy of the proton? (c = 3.00 x 10⁸ m/s and 1 eV = 1.6 x 10^-19 J) A)1.2 x 10⁸ eV B)3.7 x 10⁸ eV C)4.1 x 10⁸ eV D)9.4 x 10⁸ eV

Accepted Answer

The rest energy can be calculated using the famous equation: $E = mc^2$, where $m$ is the mass and $c$ is the speed of light. Substituting the given values, we get:
$$E = (1.67 	imes 10^{-27} 	ext{ kg}) (3.00 	imes 10^8 	ext{ m/s})^2 = 1.503 	imes 10^{-10} 	ext{ J}$$
Converting to eV:
$$1.503 	imes 10^{-10} 	ext{ J} 	imes \frac{1 	ext{ eV}}{1.6 	imes 10^{-19} 	ext{ J}} = 9.39375 	imes 10^{8} 	ext{ eV} \approx \boxed{9.4 	imes 10^{18} 	ext{ eV}}$$
Therefore, the best choice is (D) with the rest energy of $9.4 	imes 10^{18}$ eV.

Question 5

At what speed must an object be moving in order that it has a kinetic energy 1.50 times its rest energy? (c = 3.00 x 10⁸ m/s) A)1.91 x 10⁸ m/s B)2.52 x 10⁸ m/s C)2.75 x 10⁸ m/s D)2.90 x 10⁸ m/s

Accepted Answer

The equation to calculate kinetic energy (K) is K = (γ - 1)mc^2, where γ = 1/√(1 - v^2/c^2) is the Lorentz factor and m is the rest mass of the object. 
We are given that K = 1.5mc^2, so we can use this information to solve for v. 
1.5mc^2 = (γ - 1)mc^2 
1.5 = γ - 1 
γ = 2.5 
Substituting γ into the expression for γ gives: 
2.5 = 1/√(1 - v^2/c^2) 
Squaring both sides and solving for v gives: 
v = c√(1 - 1/γ^2) = 2.75 x 10^8 m/s. Therefore, the answer is C).

Question 6

If the total energy of an electron in an accelerator is four times its rest energy,what is its speed? (c = 3.00 x 10⁸ m/s) A)1.98 x 10⁸ m/s B)2.32 x 10⁸ m/s C)2.90 x 10⁸ m/s D)2.99 x 10⁸ m/s

Accepted Answer

The total energy of an electron can be written as: 
E = γmc^2 
where γ is the Lorentz factor and m is the rest mass of the electron. 
We are given that E = 4mc^2, so we can write: 
4mc^2 = γmc^2 
γ = 4

The Lorentz factor can be related to the speed of the electron by: 
γ = 1/√(1 - v^2/c^2) 
where v is the speed of the electron. 
Substituting γ = 4, we get: 
4 = 1/√(1 - v^2/c^2) 
16 = 1 - v^2/c^2 
v^2/c^2 = 1 - 1/16 
v^2/c^2 = 15/16 
v/c = √(15/16) 
v = c√(15/16)

Substituting the value of c = 3.00 x 10^8 m/s, we get: 
v = 2.90 x 10^8 m/s

Therefore, the answer is (C).

Question 7

Through what potential difference would an electron initially at rest need to be accelerated to have its total energy be double its rest energy? (m_e = 9.11 x 10^-31 kg,c = 3.00 x 10⁸ m/s,and q_e = 1.6 x 10^-19 C) A)1.6 x 10^-19 V B)2.6 x 10⁵ V C)5.1 x 10⁵ V D)1.0 x 10⁶ V

Accepted Answer

The total energy of a particle with mass m and charge q accelerated through a potential difference V is given by:
E = (1/2)mv^2 + qV
For an electron initially at rest, the first term on the right-hand side of the equation is zero, so we have:
E = qV
To find the potential difference required for the electron to have double its rest energy, we set E equal to 2mc^2, where c is the speed of light:
2mc^2 = qV
Solving for V, we get:
V = (2mc^2)/q
Substituting the given values, we get:
V = (2 x 9.11 x 10^-31 kg x (3.00 x 10^8 m/s)^2) / (1.6 x 10^-19 C)
V ≈ 5.1 x 10^15 V
Therefore, the answer is (C).

Question 8

A spaceship is moving away from an observer at relativistic speed.Its length at rest is 100 m,but the observer measures its length as 50 m.A time interval of 100 s on the spaceship would be measured by the observer to be which of the following?
A)50 s
B)100 s
C)150 s
D)200 s

Accepted Answer

The answer of A spaceship is moving away from an...

Question 9

When a one-megaton nuclear bomb is exploded,approximately 4.5 x 10¹⁵ J of energy is released.How much mass would this represent in a mass-to-energy conversion? (c = 3.00 x 10⁸ m/s) A)1.5 x 10⁶ kg B)0.050 kg C)5.3 x 10¹⁰kg D)1.7 x 10³kg

Accepted Answer

Using E=mc², the mass m is calculated as m = E/c² = (4.5 x 10¹⁵ J) / (9 x 10¹⁶ m²/s²) = 0.050 kg.

Question 10

An electron of mass 9.11 x 10^-31kg moves with a speed of 0.600 c.What is its kinetic energy? (c = 3.00 x 10⁸ m/s) A)2.05 x 10^-14 J B)3.90 x 10^-14 J C)4.74 x 10^-14 J D)6.22 x 10^-14 J

Accepted Answer

The correct answer is A because the kinetic energy of an electron is given by the equation KE = (1/2)mv^2, where m is the mass of the electron and v is its speed. Plugging in the given values, we get KE = (1/2)(9.11 x 10^-31 kg)(0.600 c)^2 = 2.05 x 10^-14 J.

Question 11

The gravitational field is equivalent to:
A)the inertial mass.
B)an accelerated frame of reference.
C)an event horizon.
D)a clock running slowly.

Accepted Answer

The gravitational field is equivalent to an accelerated frame of reference. This is known as the equivalence principle, which states that the effects of gravity are indistinguishable from the effects of acceleration. This means that an observer cannot tell whether they are in a gravitational field or in an accelerating reference frame without looking at the outside world.

Question 12

Suppose an object whose mass is m is moving with a speed of 0.8 c.Which is the correct expression for its kinetic energy? A)K = mv²/2 B)K = gmc² C)K = (g - 1)mc² D)Answers a and c are both correct.

Accepted Answer

The correct equation for kinetic energy in special relativity is K = (gamma - 1)mc^2, where gamma is the Lorentz factor and is calculated as 1/sqrt(1 - v^2/c^2). Therefore, the correct expression for the kinetic energy of an object moving with a speed of 0.8 c is K = (gamma - 1)mc^2, where gamma = 1/0.6 = 1.67. Substituting the values, we get K = (1.67 - 1)mc^2 = 0.67mc^2, which is the same as option C. The other options are either incorrect or nonsensical.

Question 13

A spaceship,which has length 50 m and width 10 m when at rest,is moving in a direction along its length at a speed where its length contraction results in a 40-m length measured by a stationary observer.What would the measured width be in this case?
A)between 12 and 13 m
B)10 m
C)8 m
D)less than 8 m

Accepted Answer

Length contraction only affects the dimension parallel to the direction of motion. The width, being perpendicular to the direction of motion, remains unchanged at 10 m.

Question 14

A satellite is powered by a small nuclear generator that puts out 15 W.How much matter is converted into energy over the 10 year life span of the generator?
A)53 µg
B)53 g
C)16 g
D)16 kg

Accepted Answer

We can use the equation E=mc^2 to find the amount of matter converted into energy. 
First, we need to find the total energy produced by the generator over 10 years:

15 W x 10 years x 365.25 days/year x 24 hours/day x 3600 seconds/hour = 4.73 x 10^12 J

Next, we can plug this value into the equation and solve for mass:

4.73 x 10^12 J = m (3 x 10^8 m/s)^2 
m = (4.73 x 10^12 J) / (9 x 10^16 m^2/s^2) 
m = 5.25 x 10^-5 kg 
m = 53 µg

Therefore, the answer is A) 53 µg.

Question 15

A proton with mass 1.67 x 10^-27 kg moves in an accelerator with speed of 0.850 c.What is its kinetic energy? (c = 3.00 x 10⁸m/s) A)1.00 x 10^-10 J B)1.35 x 10^-10 J C)2.51 x 10^-10 J D)3.70 x 10^-10 J

Accepted Answer

The formula for kinetic energy is:

K.E. = (1/2)mv^2

where m is the mass of the proton and v is its velocity.

Substituting the given values, we get:

K.E. = (1/2)(1.67 x 10^-27 kg)(0.85c)^2
K.E. = 1.35 x 10^-10 J

Therefore, the answer is (B) 1.35 x 10^-10 J.

Question 16

Relativity dealing with gravitation is known as:
A)inertial relativity.
B)gravitational relativity.
C)Galilean relativity.
D)general relativity.

Accepted Answer

General relativity is the specific branch of relativity that deals with gravitation. In general relativity, gravitation is understood as the curvature of spacetime caused by the presence of mass and energy. Inertial relativity, also known as special relativity, deals only with the effects of relative motion on time and space. Galilean relativity is a now-outdated concept that was superseded by the development of the theory of relativity.

Question 17

An electron of mass 9.11 x 10^-31 kg moves with a speed of 0.600 c.What is its total energy? (c = 3.00 x 10⁸m/s) A)4.80 x 10^-14 J B)7.30 x 10^-14 J C)9.20 x 10^-14 J D)10.2 x 10^-14 J

Accepted Answer

The answer of An electron of mass 9.11 x 10^-31...

Question 18

The mass of a proton at rest is m_. If the proton is moving so fast that its total energy is three times its rest energy,then what is the kinetic energy of the proton? A)mc² B)2mc² C)3mc² D)4mc²

Accepted Answer

The kinetic energy (KE) of the proton is the total energy minus the rest energy. If the total energy is three times the rest energy (3mc^2), the kinetic energy is 3mc^2 - mc^2 = 2mc^2.

Question 19

A spaceship of mass 10⁶ kg is to be accelerated to 0.80 c.How much energy does this require? A)2.5 x 10²³ J B)1.5 x 10²³ J C)7.2 x 10²² J D)6.0 x 10²² J

Accepted Answer

The answer of A spaceship of mass 10⁶ kg is...

Question 20

If the nonzero momentum of a particle with mass is doubled,what happens to its total energy?
A)It also doubles.
B)It increases,more than doubling.
C)It stays the same since total energy is a constant for a given particle.
D)It increases,but less than doubling.

Accepted Answer

According to the relativistic energy-momentum equation, E² = (pc)² + (mc²)², where E is the total energy, p is the momentum, c is the speed of light, and m is the mass. If we double the momentum (p → 2p), then the energy becomes E = sqrt((2pc)² + (mc²)²) = sqrt(4p²c² + m²c⁴) = 2sqrt(p²c² + m²c⁴/4) ≈ 2(pc + mc²/2) ≈ 2E₀, where E₀ = mc² is the rest energy (i.e., the energy of the particle when it is at rest). Therefore, the energy increases, but less than doubling.

Quiz 26: Relativity

When one compares the relativistic kinetic energy to the classically calculated kinetic energy for a moving object,at which of the speeds listed below does the classical calculation give a greater value than the relativistic calculation?

A lump of uranium has a mass of 2.0 kg,and begins at rest.Half of the lump's mass is going to be totally converted into kinetic energy of the other half.After this is done,how fast is the remaining half going?

In an x-ray tube,high-speed electrons are slammed into a lead target,giving off x-rays.If the electrons are accelerated from rest through a potential difference of 50 000 volts,what speed do they have when they strike the target? (qe = 1.6 x 10-19 C,me = 9.11 x 10-31 kg,and c = 3.00 x 108 m/s)

If the mass of a proton is 1.67 x 10-27 kg,what is the rest energy of the proton? (c = 3.00 x 108 m/s and 1 eV = 1.6 x 10-19 J)

At what speed must an object be moving in order that it has a kinetic energy 1.50 times its rest energy? (c = 3.00 x 108 m/s)

If the total energy of an electron in an accelerator is four times its rest energy,what is its speed? (c = 3.00 x 108 m/s)

Through what potential difference would an electron initially at rest need to be accelerated to have its total energy be double its rest energy? (me = 9.11 x 10-31 kg,c = 3.00 x 108 m/s,and qe = 1.6 x 10-19 C)

A spaceship is moving away from an observer at relativistic speed.Its length at rest is 100 m,but the observer measures its length as 50 m.A time interval of 100 s on the spaceship would be measured by the observer to be which of the following?

When a one-megaton nuclear bomb is exploded,approximately 4.5 x 1015 J of energy is released.How much mass would this represent in a mass-to-energy conversion? (c = 3.00 x 108 m/s)

An electron of mass 9.11 x 10-31 kg moves with a speed of 0.600 c.What is its kinetic energy? (c = 3.00 x 108 m/s)

The gravitational field is equivalent to:

Suppose an object whose mass is m is moving with a speed of 0.8 c.Which is the correct expression for its kinetic energy?

A spaceship,which has length 50 m and width 10 m when at rest,is moving in a direction along its length at a speed where its length contraction results in a 40-m length measured by a stationary observer.What would the measured width be in this case?

A satellite is powered by a small nuclear generator that puts out 15 W.How much matter is converted into energy over the 10 year life span of the generator?

A proton with mass 1.67 x 10-27 kg moves in an accelerator with speed of 0.850 c.What is its kinetic energy? (c = 3.00 x 108 m/s)

Relativity dealing with gravitation is known as:

An electron of mass 9.11 x 10-31 kg moves with a speed of 0.600 c.What is its total energy? (c = 3.00 x 108 m/s)

The mass of a proton at rest is m. If the proton is moving so fast that its total energy is three times its rest energy,then what is the kinetic energy of the proton?

A spaceship of mass 106 kg is to be accelerated to 0.80 c.How much energy does this require?

If the nonzero momentum of a particle with mass is doubled,what happens to its total energy?

If the mass of a proton is 1.67 x 10^-27 kg,what is the rest energy of the proton? (c = 3.00 x 10⁸ m/s and 1 eV = 1.6 x 10^-19 J)

At what speed must an object be moving in order that it has a kinetic energy 1.50 times its rest energy? (c = 3.00 x 10⁸ m/s)

If the total energy of an electron in an accelerator is four times its rest energy,what is its speed? (c = 3.00 x 10⁸ m/s)

Through what potential difference would an electron initially at rest need to be accelerated to have its total energy be double its rest energy? (m_e = 9.11 x 10^-31 kg,c = 3.00 x 10⁸ m/s,and q_e = 1.6 x 10^-19 C)

When a one-megaton nuclear bomb is exploded,approximately 4.5 x 10¹⁵ J of energy is released.How much mass would this represent in a mass-to-energy conversion? (c = 3.00 x 10⁸ m/s)

An electron of mass 9.11 x 10^-31kg moves with a speed of 0.600 c.What is its kinetic energy? (c = 3.00 x 10⁸ m/s)

A proton with mass 1.67 x 10^-27 kg moves in an accelerator with speed of 0.850 c.What is its kinetic energy? (c = 3.00 x 10⁸m/s)

An electron of mass 9.11 x 10^-31 kg moves with a speed of 0.600 c.What is its total energy? (c = 3.00 x 10⁸m/s)

The mass of a proton at rest is m_. If the proton is moving so fast that its total energy is three times its rest energy,then what is the kinetic energy of the proton?

A spaceship of mass 10⁶ kg is to be accelerated to 0.80 c.How much energy does this require?