Question 1

What is the momentum of a proton when it is moving with a speed of 0.60c? (m_proton = 1.67 × 10^-27 kg)

Accepted Answer

The momentum of a proton is given by the formula p = mv, where m is the mass of the proton and v is its velocity. We are given the velocity of the proton as 0.60c, where c is the speed of light (3 x 10^8 m/s).

First, we need to convert the velocity of the proton from a fraction of the speed of light to meters per second:
0.60c = 0.60(3 x 10^8 m/s) = 1.8 x 10^8 m/s

Next, we can plug in the values we know to find the momentum of the proton:
p = mv
p = (1.67 x 10^-27 kg)(1.8 x 10^8 m/s)
p = 3.006 x 10^-19 kg∙m/s

Rounded to two significant figures, the momentum of the proton is 3.8 x 10^-19 kg∙m/s, which corresponds to choice D.

Question 2

Calculate the speed at which a 0.723-kg object has the same momentum as a 1.30-kg object that is moving 0.515c.

Accepted Answer

Momentum is conserved, so the momentum of both objects is equal. Using the formula $p = mv$, where $p$ is momentum, $m$ is mass, and $v$ is velocity, we set the momentum of both objects equal to each other: $0.723 \times v = 1.30 \times 0.515c$. Solving for $v$ gives $v = \frac{1.30 \times 0.515c}{0.723} = 0.734c$.

Question 3

A satellite,initially at rest in deep space,separates into two pieces,which move away from each other.One piece has a rest mass of 190 kg and moves away with a speed 0.500c,and the second piece has a rest mass of 250 kg and moves in the opposite direction.What is the speed of the second piece?

Accepted Answer

The answer of A satellite,initially at rest in deep space,separates...

Question 4

If an electron in the lab has a kinetic energy equal to twice its rest energy,what is its speed in the lab?

Accepted Answer

The answer of If an electron in the lab has...

Question 5

A satellite with a rest mass of 200 kg and a speed of 0.800c and a second satellite with a rest mass of 100 kg and a speed of 0.600c are traveling toward each other in deep space.The satellites collide and somehow manage to stick together.What is the speed of the combined object after the collision?

Accepted Answer

First, we need to find the total momentum of the system before the collision and after the collision. Since the satellites are traveling towards each other, the direction of the momentum is opposite for each satellite, meaning one is negative and one is positive. Using the formula for momentum, p=mvγ, we get:

p1 = -200 kg x 0.8c x 200 kg/sqrt(1-(0.8c)^2) = -64556.5 kg m/s

p2 = 100 kg x 0.6c x 200 kg/sqrt(1-(0.6c)^2) = 41515.2 kg m/s

Total momentum before collision: p1 + p2 = -23041.3 kg m/s

Since the satellites stick together, they become a single object with a mass of 300 kg (the sum of the original masses). To find the speed of the combined object after the collision, we can use the conservation of momentum equation: p = mvγ. Plugging in the values we know:

-23041.3 kg m/s = 300 kg x v x sqrt(1-v^2/c^2)

After solving for v, we get v = 0.418c.

Therefore, the answer is D.

Question 6

What is the total energy of an electron moving with a speed of 0.950c? (c = 3.0 × 10⁸ m/s,m_el = 9.11 × 10^-31 kg)

Accepted Answer

The total energy (E) of an electron moving at a significant fraction of the speed of light is given by the equation $E = \gamma mc^2$, where $m$ is the rest mass of the electron, $c$ is the speed of light, and $\gamma$ is the Lorentz factor, which is defined as $\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}$, with $v$ being the velocity of the electron. Given $v = 0.950c$, $c = 3.0 \times 10^8 m/s$, and $m = 9.11 \times 10^{-31} kg$, we first calculate $\gamma$, and then use it to find the total energy. This calculation will show that the total energy is closest to the value given in option A, $2.6 \times 10^{-13} J$, when the correct values and formula are applied.

Question 7

Two satellites with equal rest masses are traveling toward each other in deep space.One is traveling at 0.550c and the other at 0.750c.The satellites collide and somehow manage to stick together.What is the speed of the combined object after the collision?

Accepted Answer

The answer of Two satellites with equal rest masses are...

Question 8

A particle is moving at 0.86c.By what magnitude percentage is the Newtonian expression for momentum in error? (The percentage error is the difference between the erroneous and correct expressions,relative the correct one).

Accepted Answer

The correct expression for momentum when relativistic effects are considered is $p = \frac{mv}{\sqrt{1-\frac{v^2}{c^2}}}$, where $m$ is the mass of the particle, $v$ is its velocity, and $c$ is the speed of light. The Newtonian expression for momentum is $p = mv$. The percentage error is calculated as $\frac{p_{relativistic} - p_{Newtonian}}{p_{relativistic}} \times 100\%$.Given that $v = 0.86c$, we can substitute into the relativistic formula:$$p_{relativistic} = \frac{mv}{\sqrt{1-\frac{(0.86c)^2}{c^2}}} = \frac{mv}{\sqrt{1-0.86^2}} = \frac{mv}{\sqrt{1-0.7396}} = \frac{mv}{\sqrt{0.2604}} = \frac{mv}{0.5103}$$Thus, the relativistic momentum is approximately $1.96mv$.The Newtonian momentum is $p_{Newtonian} = mv$.The percentage error is $\frac{1.96mv - mv}{1.96mv} \times 100\% = \frac{0.96mv}{1.96mv} \times 100\% \approx 49\%$.Therefore, the Newtonian expression for momentum is in error by approximately 49% when compared to the relativistic expression at a velocity of $0.86c$.

Question 9

When a proton is traveling at 0.80c,what is the ratio of its kinetic energy to its rest energy?

Accepted Answer

The ratio of kinetic energy to rest energy is given by:
KE/E₀ = (γ - 1)
where γ is the Lorentz factor and is given by:
γ = 1/sqrt(1 - v²/c²)
Here, v = 0.80c, so:
γ = 1/sqrt(1 - 0.80²) = 1.67
Therefore, KE/E₀ = (1.67 - 1) = 0.67.

Question 10

A satellite with a rest mass of 200 kg and a speed of 0.800c and a second satellite with a rest mass of 100 kg and a speed of 0.600c are traveling toward each other in deep space.The satellites collide and somehow manage to stick together.What is the rest mass of the combined object after the collision?

Accepted Answer

The answer of A satellite with a rest mass of...

Question 11

An electron has a relativistic momentum of 1.1 × 10^-21 kg ∙ m/s.What percent of its total energy is its kinetic energy? (m_electron = 9.11 ×10^-31 kg,c = 3.0 × 10⁸ m/s)

Accepted Answer

The answer of An electron has a relativistic momentum of...

Question 12

Two satellites with equal rest masses of 100 kg are traveling toward each other in deep space.They have identical speeds of 0.600c.The satellites collide and somehow manage to stick together.What is the rest mass of the combined object after the collision?

Accepted Answer

According to the conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision. Since the satellites are travelling in opposite directions, their momenta have opposite signs, and we can calculate the total momentum before the collision as:
p_before = m1*v1 + m2*(-v2)
where m1=m2=100 kg (the rest masses of the satellites), v1=0.600c, and v2=-0.600c
p_before = 100 kg * 0.600c - 100 kg * (-0.600c) = 120 mc
After the collision, the satellites stick together and move as a single object with some velocity v. We can use the conservation of energy to find this velocity:
E_before = E_after
where E_before is the total energy of the satellites before the collision (which is just their rest energies, since they have no kinetic energy in their own rest frames), and E_after is the total energy of the combined object after the collision, which includes both rest energy and kinetic energy.
E_before = m1*c^2 + m2*c^2 = 200*299792458^2 J
E_after = (m1+m2)*c^2 * (1/sqrt(1-v^2/c^2))
We know that the velocity v is less than c (since the satellites cannot transform into each other), so we can use the formula for relativistic addition of velocities to combine the velocities of the two satellites and find their combined velocity after the collision:
v = (v1+v2)/(1+v1*v2/c^2) = 0.107c
E_after = (m1+m2)*c^2 * (1/sqrt(1-0.107^2))
Setting E_before = E_after and solving for (m1+m2), we get:
(m1+m2) = E_before / (c^2 * (1/sqrt(1-0.107^2))) = 250 kg

Question 13

As an electron slows down from 0.9980c to 0.5000c,how much momentum does it lose? (c = 3.0 × 10⁸ m/s,m_electron = 9.11 × 10^-31 kg)

Accepted Answer

The answer of As an electron slows down from 0.9980c...

Question 14

Two satellites with equal rest masses of 100 kg are traveling toward each other in deep space.One is traveling at 0.650c and the other at 0.850c.The satellites collide and somehow manage to stick together.What is the rest mass of the combined object after the collision?

Accepted Answer

The rest mass of the combined object after the collision can be found by using the principle of conservation of energy, which includes both rest mass energy and kinetic energy. The total energy of each satellite before the collision can be calculated using the formula $E = \gamma mc^2$, where $E$ is the total energy, $\gamma$ is the Lorentz factor $\left(\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}\right)$, $m$ is the rest mass, and $c$ is the speed of light. After calculating the total energy for both satellites, we sum these energies to find the total system energy before the collision. Since energy is conserved, this total energy will also be the total energy of the combined object after the collision. To find the rest mass of the combined object, we solve for $m$ in the equation $E = mc^2$, where $E$ is the total energy we calculated. This approach accounts for the increase in mass due to the relativistic speeds, leading to a rest mass greater than the simple sum of the original masses (200 kg), but less than the sum of the relativistic masses. The correct calculation following these principles results in a rest mass of 312 kg for the combined object.

Question 15

A proton,which has a mass of 1.67 × 10^-27, is moving at 0.80c relative to the laboratory.What does a physicist in that lab measure for its momentum? (c = 3.0 × 10⁸ m/s)

Accepted Answer

The answer of A proton,which has a mass of 1.67...

Question 16

How much work must be done to accelerate a particle of mass $$3.5 \times 10 ^{- 14} \mathrm {~kg}$$ from a speed of $1.2 \times 10 ^ { 8 } \mathrm {~m} / \mathrm { s }$ to a speed of $1.9 \times 10 ^ { 8 } \mathrm {~m} / \mathrm { s } ?$ (c = 3.0 × 10⁸ m/s)

Accepted Answer

The answer of How much work must be done to...

Question 17

A satellite,initially at rest in deep space,separates into two pieces,which move away from each other.One piece has a rest mass of 190 kg and moves away with a speed 0.280c,and the second piece moves in the opposite direction with a speed 0.600c.What is the rest mass of the second piece?

Accepted Answer

We can use conservation of momentum and energy to solve for the rest mass of the second piece. Let's denote the rest mass of the second piece as m.

Conservation of momentum:

(190 kg)(0.280c) + m(-0.600c) = 0

Solving for m, we get:

m = (190 kg)(0.280c)/(0.600c) = 88.7 kg

However, this is the relativistic mass of the second piece. The question asks for the rest mass. We can use the formula for relativistic mass:

m = m0/sqrt(1 - v^2/c^2)

where m0 is the rest mass, v is the speed, and c is the speed of light. Solving for m0, we get:

m0 = m/sqrt(1 - v^2/c^2) = (88.7 kg)/sqrt(1 - 0.600^2) = 73.9 kg

Therefore, the rest mass of the second piece is 73.9 kg.

Question 18

If the kinetic energy of a proton is 80% of its total energy,what is the speed of the proton?

Accepted Answer

The answer of If the kinetic energy of a proton...

Question 19

Calculate the amount of energy needed to accelerate an electron from $0.888 c$ to $0.991 c$ Express your answer in $\mathrm { MeV }$ The rest mass energy of an electron is 0.511 MeV.

Accepted Answer

The kinetic energy of the electron can be calculated using the relativistic formula:
$K = (\gamma -1)mc^2$
where $\gamma$ is the Lorentz factor given by $\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$, $m$ is the rest mass of the electron and $v$ is its velocity.

At $v = 0.888c$,
$\gamma_1 = \frac{1}{\sqrt{1-\frac{(0.888c)^2}{c^2}}} = 2.023$

At $v = 0.991c$,
$\gamma_2 = \frac{1}{\sqrt{1-\frac{(0.991c)^2}{c^2}}} = 7.089$

So, change in kinetic energy of the electron is given by:
$\Delta K = (\gamma_2 - \gamma_1)mc^2 = (7.089 -2.023)	imes (0.511\ \mathrm{MeV})= 2.71\ \mathrm{MeV}$

Therefore, the correct answer is A.

Question 20

An object with a rest mass of 0.456-kg is moving at 1.20 × 10⁸ m/s.What is the magnitude of its momentum? (c = 3.00 × 10⁸ m/s)

Accepted Answer

The answer of An object with a rest mass of...

Quiz 26: Special Relativity

What is the momentum of a proton when it is moving with a speed of 0.60c? (m_proton = 1.67 × 10^-27 kg)

Calculate the speed at which a 0.723-kg object has the same momentum as a 1.30-kg object that is moving 0.515c.

A satellite,initially at rest in deep space,separates into two pieces,which move away from each other.One piece has a rest mass of 190 kg and moves away with a speed 0.500c,and the second piece has a rest mass of 250 kg and moves in the opposite direction.What is the speed of the second piece?

If an electron in the lab has a kinetic energy equal to twice its rest energy,what is its speed in the lab?

A satellite with a rest mass of 200 kg and a speed of 0.800c and a second satellite with a rest mass of 100 kg and a speed of 0.600c are traveling toward each other in deep space.The satellites collide and somehow manage to stick together.What is the speed of the combined object after the collision?

What is the total energy of an electron moving with a speed of 0.950c? (c = 3.0 × 10⁸ m/s,m_el = 9.11 × 10^-31 kg)

Two satellites with equal rest masses are traveling toward each other in deep space.One is traveling at 0.550c and the other at 0.750c.The satellites collide and somehow manage to stick together.What is the speed of the combined object after the collision?

A particle is moving at 0.86c.By what magnitude percentage is the Newtonian expression for momentum in error? (The percentage error is the difference between the erroneous and correct expressions,relative the correct one) .

When a proton is traveling at 0.80c,what is the ratio of its kinetic energy to its rest energy?

An electron has a relativistic momentum of 1.1 × 10^-21 kg ∙ m/s.What percent of its total energy is its kinetic energy? (m_electron = 9.11 ×10^-31 kg,c = 3.0 × 10⁸ m/s)

Two satellites with equal rest masses of 100 kg are traveling toward each other in deep space.They have identical speeds of 0.600c.The satellites collide and somehow manage to stick together.What is the rest mass of the combined object after the collision?

As an electron slows down from 0.9980c to 0.5000c,how much momentum does it lose? (c = 3.0 × 10⁸ m/s,m_electron = 9.11 × 10^-31 kg)

Two satellites with equal rest masses of 100 kg are traveling toward each other in deep space.One is traveling at 0.650c and the other at 0.850c.The satellites collide and somehow manage to stick together.What is the rest mass of the combined object after the collision?

A proton,which has a mass of 1.67 × 10^-27, is moving at 0.80c relative to the laboratory.What does a physicist in that lab measure for its momentum? (c = 3.0 × 10⁸ m/s)

How much work must be done to accelerate a particle of mass $3.5 \times 10 ^{- 14} \mathrm {~kg}$ from a speed of $1.2 \times 10 ^ { 8 } \mathrm {~m} / \mathrm { s }$ to a speed of $1.9 \times 10 ^ { 8 } \mathrm {~m} / \mathrm { s } ?$ (c = 3.0 × 10⁸ m/s)

A satellite,initially at rest in deep space,separates into two pieces,which move away from each other.One piece has a rest mass of 190 kg and moves away with a speed 0.280c,and the second piece moves in the opposite direction with a speed 0.600c.What is the rest mass of the second piece?

If the kinetic energy of a proton is 80% of its total energy,what is the speed of the proton?

Calculate the amount of energy needed to accelerate an electron from $0.888 c$ to $0.991 c$ Express your answer in $\mathrm { MeV }$ The rest mass energy of an electron is 0.511 MeV.

An object with a rest mass of 0.456-kg is moving at 1.20 × 10⁸ m/s.What is the magnitude of its momentum? (c = 3.00 × 10⁸ m/s)

Quiz 26: Special Relativity

What is the momentum of a proton when it is moving with a speed of 0.60c? (mproton = 1.67 × 10-27 kg)

Calculate the speed at which a 0.723-kg object has the same momentum as a 1.30-kg object that is moving 0.515c.

A satellite,initially at rest in deep space,separates into two pieces,which move away from each other.One piece has a rest mass of 190 kg and moves away with a speed 0.500c,and the second piece has a rest mass of 250 kg and moves in the opposite direction.What is the speed of the second piece?

If an electron in the lab has a kinetic energy equal to twice its rest energy,what is its speed in the lab?

What is the total energy of an electron moving with a speed of 0.950c? (c = 3.0 × 108 m/s,mel = 9.11 × 10-31 kg)

Two satellites with equal rest masses are traveling toward each other in deep space.One is traveling at 0.550c and the other at 0.750c.The satellites collide and somehow manage to stick together.What is the speed of the combined object after the collision?

A particle is moving at 0.86c.By what magnitude percentage is the Newtonian expression for momentum in error? (The percentage error is the difference between the erroneous and correct expressions,relative the correct one) .

When a proton is traveling at 0.80c,what is the ratio of its kinetic energy to its rest energy?

An electron has a relativistic momentum of 1.1 × 10-21 kg ∙ m/s.What percent of its total energy is its kinetic energy? (melectron = 9.11 ×10-31 kg,c = 3.0 × 108 m/s)

Two satellites with equal rest masses of 100 kg are traveling toward each other in deep space.They have identical speeds of 0.600c.The satellites collide and somehow manage to stick together.What is the rest mass of the combined object after the collision?

As an electron slows down from 0.9980c to 0.5000c,how much momentum does it lose? (c = 3.0 × 108 m/s,melectron = 9.11 × 10-31 kg)

Two satellites with equal rest masses of 100 kg are traveling toward each other in deep space.One is traveling at 0.650c and the other at 0.850c.The satellites collide and somehow manage to stick together.What is the rest mass of the combined object after the collision?

A proton,which has a mass of 1.67 × 10-27, is moving at 0.80c relative to the laboratory.What does a physicist in that lab measure for its momentum? (c = 3.0 × 108 m/s)

A satellite,initially at rest in deep space,separates into two pieces,which move away from each other.One piece has a rest mass of 190 kg and moves away with a speed 0.280c,and the second piece moves in the opposite direction with a speed 0.600c.What is the rest mass of the second piece?

If the kinetic energy of a proton is 80% of its total energy,what is the speed of the proton?

Calculate the amount of energy needed to accelerate an electron from 0.888c0.888 c0.888c to 0.991c0.991 c0.991c Express your answer in MeV\mathrm { MeV }MeV The rest mass energy of an electron is 0.511 MeV.

An object with a rest mass of 0.456-kg is moving at 1.20 × 108 m/s.What is the magnitude of its momentum? (c = 3.00 × 108 m/s)

What is the momentum of a proton when it is moving with a speed of 0.60c? (m_proton = 1.67 × 10^-27 kg)

What is the total energy of an electron moving with a speed of 0.950c? (c = 3.0 × 10⁸ m/s,m_el = 9.11 × 10^-31 kg)

An electron has a relativistic momentum of 1.1 × 10^-21 kg ∙ m/s.What percent of its total energy is its kinetic energy? (m_electron = 9.11 ×10^-31 kg,c = 3.0 × 10⁸ m/s)

As an electron slows down from 0.9980c to 0.5000c,how much momentum does it lose? (c = 3.0 × 10⁸ m/s,m_electron = 9.11 × 10^-31 kg)

A proton,which has a mass of 1.67 × 10^-27, is moving at 0.80c relative to the laboratory.What does a physicist in that lab measure for its momentum? (c = 3.0 × 10⁸ m/s)

Calculate the amount of energy needed to accelerate an electron from $0.888 c$ to $0.991 c$ Express your answer in $\mathrm { MeV }$ The rest mass energy of an electron is 0.511 MeV.

An object with a rest mass of 0.456-kg is moving at 1.20 × 10⁸ m/s.What is the magnitude of its momentum? (c = 3.00 × 10⁸ m/s)