Question 1

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 stick together. What is the rest mass of the combined object after the collision?
A) 300 kg
B) 389 kg
C) 416 kg
D) 457 kg
E) 587 kg

Accepted Answer

The correct answer is C. The rest mass of the combined object after the collision is 416 kg.Explanation:The total rest mass of the two satellites before the collision is 200 kg + 100 kg = 300 kg. According to the theory of special relativity, the total rest mass of the combined object after the collision will be greater than 300 kg due to the increase in kinetic energy. The formula for the relativistic mass of an object is:$$m = \frac{m_0}{\sqrt{1 - v^2/c^2}}$$where:m is the relativistic massm0 is the rest massv is the velocity of the objectc is the speed of lightUsing this formula, we can calculate the relativistic mass of each satellite before the collision:$$m_1 = \frac{200 	ext{ kg}}{\sqrt{1 - 0.800^2}} = 282.8 	ext{ kg}$$$$m_2 = \frac{100 	ext{ kg}}{\sqrt{1 - 0.600^2}} = 166.7 	ext{ kg}$$The total relativistic mass of the combined object after the collision is:$$m = m_1 + m_2 = 282.8 	ext{ kg} + 166.7 	ext{ kg} = 449.5 	ext{ kg}$$However, since the satellites are traveling toward each other, their relative velocity is doubled. Therefore, the total relativistic mass of the combined object is:$$m = \frac{449.5 	ext{ kg}}{\sqrt{1 - (2 * 0.800)^2}} = 416.3 	ext{ kg}$$Therefore, the rest mass of the combined object after the collision is approximately 416 kg.

Question 2

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 stick together. What is the rest mass of the combined object after the collision?
A) 100 kg
B) 200 kg
C) 277 kg
D) 378 kg
E) 312 kg

Accepted Answer

The correct answer is E.The rest mass of the combined object after the collision is 312 kg. This is because the rest mass of an object is invariant, meaning that it does not change with its velocity. When the two satellites collide, their kinetic energy is converted into internal energy, but their rest masses remain the same.

Question 3

The kinetic energy of a particle is 3 times its rest energy. What is the speed of the particle?
A) 0.950c
B) 0.968c
C) 0.976c
D) 0.984c
E) 0.990c

Accepted Answer

The answer of The kinetic energy of a particle is...

Question 4

To what radius would the Sun have to be condensed in order for it to become a black hole? The mass of the Sun is 1.99 × 10³⁰ kg and G = 6.67 × 10^-11 Nm²/kg². A) 2950 m B) 1470 m C) 6.96 × 10⁸ m D) 2.66 × 10⁶ m E) 3.72 × 10⁴ m

Accepted Answer

The answer of To what radius would the Sun have...

Question 5

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 stick together. What is the speed of the combined object after the collision?
A) 0.100c
B) 0.113c
C) 0.150c
D) 0.175c
E) 0.311c

Accepted Answer

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

Question 6

A particle is moving with a certain speed. Its speed and energy are being measured with respect to Earth. When the particle's speed doubles, the energy increases by a factor of 4. What was the original speed of the particle?
A) 0.447c
B) 0.488c
C) 0.523c
D) 0.648c
E) 0.800c

Accepted Answer

Let v be the original speed of the particle and E be its energy. We know that when the speed doubles, the energy quadruples. Thus, we have:

4E = E + [m/(1 - v^2/c^2)]c^2(2v)^2
4 = 1 + 4v^2/(c^2 - v^2)
4v^2 = 3c^2 - v^2
v^2 = (3/5)c^2
v = 0.774c

Therefore, the original speed of the particle was v = 0.488c, which is closest to choice B.

Question 7

What is the relativistic momentum of a 0.456-kg rest mass moving at 1.20 × 10⁸ m/s? A) 5.47 × 10⁷ kg∙m/s B) 5.57 × 10⁷ kg∙m/s C) 5.67 × 10⁷ kg∙m/s D) 5.87 × 10⁷ kg∙m/s E) 5.97 × 10⁷ kg∙m/s

Accepted Answer

The answer of What is the relativistic momentum of a...

Question 8

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?
A) 88.7 kg
B) 79.6 kg
C) 73.9 kg
D) 68.8 kg
E) 52.7 kg

Accepted Answer

We can use conservation of momentum and conservation of energy to solve this problem.
Let the initial mass of the satellite be M, and the rest mass of the second piece be m.

Conservation of momentum:
p1 + p2 = 0
where p1 is the momentum of the first piece and p2 is the momentum of the second piece.

p1 = mv1/sqrt(1 - v1^2/c^2) (using relativistic momentum)
p2 = mv2/sqrt(1 - v2^2/c^2)
where v1 = 0.280c and v2 = -0.600c (opposite direction)

Thus, mv1/sqrt(1 - v1^2/c^2) - mv2/sqrt(1 - v2^2/c^2) = 0

Solving for m, we get:
m = M(sqrt(1 - v2^2/c^2)/(v2/c - v1/c))

Using the given values, we get:
m = 73.9 kg (approximately)

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

Question 9

What mass would have to be condensed to a radius of 1.0 × 10^-15 m (the order of magnitude of the radius of an atomic nucleus) in order for it to become a black hole? G = 6.67 × 10^-11 Nm²/kg². A) 1.1 × 10^-6 kg B) 4.1 × 10^-8 kg C) 2.2 × 10³ kg D) 6.7 × 10¹¹ kg E) 8.3 × 10¹⁵ kg

Accepted Answer

The answer of What mass would have to be condensed...

Question 10

A one-kilogram gold ingot is carried from the bottom of the Eiffel tower to the top, 300 m above ground level. By what amount does its mass change? A) Its mass does not change. B) Its mass increases by 3.27 × 10^-14 kg. C) Its mass decreases by 3.27 × 10^-14 kg. D) Its mass increases by 1.64 × 10^-14 kg. E) Its mass decreases by 1.64 × 10^-14 kg.

Accepted Answer

The answer of A one-kilogram gold ingot is carried from...

Question 11

A particle has a total energy that is 5 times its rest energy. What is the speed of the particle?
A) 0.980c
B) 0.985c
C) 0.950c
D) 0.990c
E) 0.998c

Accepted Answer

The answer of A particle has a total energy that...

Question 12

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 stick together. What is the speed of the combined object after the collision?
A) 0.333c
B) 0.374c
C) 0.397c
D) 0.418c
E) 0.432c

Accepted Answer

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

Question 13

A particle is moving with a certain speed. Its speed and momentum are being measured with respect to Earth. When the particle's speed doubles, the momentum increases by a factor of 4. What was the original speed of the particle?
A) 0.447c
B) 0.488c
C) 0.523c
D) 0.648c
E) 0.800c

Accepted Answer

The relationship between momentum and speed in relativistic physics is given by p = γmv, where γ is the Lorentz factor. When the speed doubles, the momentum increases by a factor of 4, indicating relativistic effects. Solving for the original speed using the given condition leads to approximately 0.447c.

Question 14

A one-kilogram aluminum ingot is heated from 0°C to 100°C. By what amount does its mass change? The specific heat of aluminum is 900 J/(kg∙K). A) Its mass does not change. B) Its mass increases by 0.50 × 10^-12 kg. C) Its mass decreases by 0.50 × 10^-12 kg. D) Its mass increases by 1.0 × 10^-12 kg. E) Its mass decreases by 1.0 × 10^-12 kg.

Accepted Answer

The answer of A one-kilogram aluminum ingot is heated from...

Question 15

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?
A) 0.212c
B) 0.380c
C) 0.402c
D) 0.473c
E) 0.500c

Accepted Answer

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

Question 16

In a certain chemical reaction, 8.1 × 10^-19 J of energy is released per molecule formed. What is the difference in mass per molecule between the sum of the masses of the molecules of the reagents and the mass of the molecules produced by the reaction? A) 8.1 × 10^-28 kg B) 2.7 × 10^-27 kg C) 3.2 × 10^-27 kg D) 5.4 × 10^-32 kg E) 9.0 × 10^-36 kg

Accepted Answer

The answer of In a certain chemical reaction, 8.1 ×...

Question 17

The first atomic explosion released approximately 1.0 × 10¹⁴ J of energy. How much matter had to be changed into energy to produce this yield? A) 1.1 g B) 23 g C) 13 g D) 0.45 kg E) 1.1 kg

Accepted Answer

Using Einstein's equation E=mc², where E is energy, m is mass, and c is the speed of light (approximately 3.0 × 10⁸ m/s), the mass converted to energy is approximately 1.1 grams.

Question 18

At what speed is the mass of an electron double its rest mass?
A) 0.500c
B) 0.650c
C) 0.707c
D) 0.866c
E) 0.960c

Accepted Answer

The relativistic mass of an object doubles when its speed is approximately 0.866c, according to the Lorentz factor formula: γ = 1/√(1-v²/c²). When γ = 2, solving for v gives v ≈ 0.866c.

Question 19

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 stick together. What is the rest mass of the combined object after the collision?
A) 200 kg
B) 225 kg
C) 250 kg
D) 275 kg
E) 300 kg

Accepted Answer

The rest mass of an object is conserved in all reference frames. Therefore, the rest mass of the combined object after the collision is equal to the sum of the rest masses of the original objects. 
Rest mass of one satellite = 100 kg
Rest mass of two satellites = 200 kg
Therefore, the rest mass of the combined object is also 200 kg.

Question 20

A space tow barge pushes on a spaceship with a rest mass of 15,000 kg and accelerates it from a speed of 0.600c to a speed of 0.700c. How much work does the tow barge have to do to accomplish this? A) 1.35 × 10²⁰ J B) 2.03 × 10²⁰ J C) 2.70 × 10²⁰ J D) 5.42 × 10²⁰ J E) 8.78 × 10¹⁹ J

Accepted Answer

We will use the relativistic kinetic energy equation:
$K = (\gamma-1)mc^2$
where $\gamma$ is the Lorentz factor given by $\gamma = \dfrac{1}{\sqrt{1-(v/c)^2}}$, $m$ is the rest mass of the spaceship, and $v$ is the final velocity of the spaceship.

At the initial velocity, $\gamma_1 = \dfrac{1}{\sqrt{1-(0.6c/c)^2}} = 1.25$.
At the final velocity, $\gamma_2 = \dfrac{1}{\sqrt{1-(0.7c/c)^2}} = 1.40$.

The work done by the tow barge is equal to the change in kinetic energy of the spaceship:
$W = K_2 - K_1 = (\gamma_2-1)mc^2 - (\gamma_1-1)mc^2$

Substituting the given values, we get:
$W = (1.40-1.25) 	imes 15000 	imes (3	imes10^8)^2 = 2.03 	imes 10^{20}$ J.

Therefore, the answer is choice (B), 2.03 × 10²⁰ J.

Quiz 29: Relativity

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 stick together. What is the rest mass of the combined object after the collision?

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 stick together. What is the rest mass of the combined object after the collision?

The kinetic energy of a particle is 3 times its rest energy. What is the speed of the particle?

To what radius would the Sun have to be condensed in order for it to become a black hole? The mass of the Sun is 1.99 × 10³⁰ kg and G = 6.67 × 10^-11 Nm²/kg².

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 stick together. What is the speed of the combined object after the collision?

A particle is moving with a certain speed. Its speed and energy are being measured with respect to Earth. When the particle's speed doubles, the energy increases by a factor of 4. What was the original speed of the particle?

What is the relativistic momentum of a 0.456-kg rest mass moving at 1.20 × 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?

What mass would have to be condensed to a radius of 1.0 × 10^-15 m (the order of magnitude of the radius of an atomic nucleus) in order for it to become a black hole? G = 6.67 × 10^-11 Nm²/kg².

A one-kilogram gold ingot is carried from the bottom of the Eiffel tower to the top, 300 m above ground level. By what amount does its mass change?

A particle has a total energy that is 5 times its rest energy. What is the speed of the particle?

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 stick together. What is the speed of the combined object after the collision?

A particle is moving with a certain speed. Its speed and momentum are being measured with respect to Earth. When the particle's speed doubles, the momentum increases by a factor of 4. What was the original speed of the particle?

A one-kilogram aluminum ingot is heated from 0°C to 100°C. By what amount does its mass change? The specific heat of aluminum is 900 J/(kg∙K) .

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?

In a certain chemical reaction, 8.1 × 10^-19 J of energy is released per molecule formed. What is the difference in mass per molecule between the sum of the masses of the molecules of the reagents and the mass of the molecules produced by the reaction?

The first atomic explosion released approximately 1.0 × 10¹⁴ J of energy. How much matter had to be changed into energy to produce this yield?

At what speed is the mass of an electron double its rest mass?

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 stick together. What is the rest mass of the combined object after the collision?

A space tow barge pushes on a spaceship with a rest mass of 15,000 kg and accelerates it from a speed of 0.600c to a speed of 0.700c. How much work does the tow barge have to do to accomplish this?