Okay, so I need to solve this physics problem about nuclear fission and compare the kinetic energy of the more massive fragment using classical and relativistic approaches. Let's start by understanding the problem step by step.

The problem states that an atomic nucleus of mass M is at rest with a rest-mass energy of 300 GeV. It undergoes spontaneous fission into two fragments. One fragment is twice as massive as the other. The sum of their rest masses is 99% of the initial mass M. The question is about the difference between the correct kinetic energy (T1) of the more massive fragment and the value calculated using the classical (non-relativistic) approximation.

First, let's break down the given information:

1. Initial mass of the nucleus: M. Its rest-mass energy is 300 GeV. So, using E=mc², M = 300 GeV/c². Since the problem mentions rest-mass energy, we can work in units where c=1, so mass and energy (in GeV/c² and GeV) are interchangeable.

2. After fission, two fragments: let's denote their masses as m and 2m. The sum of their rest masses is m + 2m = 3m. The problem states that this sum is 99% of M. So, 3m = 0.99M. Therefore, m = 0.99M / 3 ≈ 0.33M. Wait, but M is the initial mass. Wait, the initial rest mass is M, and the sum of the fragments' rest masses is 0.99M. So 3m = 0.99M → m = 0.33M. Then the more massive fragment is 2m = 0.66M. Wait, but the initial mass M is 300 GeV/c². So, the rest masses of the fragments are 0.33M and 0.66M. Wait, but 0.33*300 GeV = 99 GeV, and 0.66*300 GeV = 198 GeV. But the sum is 99 + 198 = 297 GeV, which is 99% of 300 GeV. That checks out.

But wait, the initial rest mass energy is 300 GeV. After fission, the total rest mass is 297 GeV, which is 99% of 300. So the rest mass of the two fragments is 297 GeV, and the remaining 3 GeV must be the kinetic energy of the fragments. Wait, but energy conservation applies here. The initial rest energy is 300 GeV. After fission, the rest energy of the fragments is 297 GeV, so the total energy (rest plus kinetic) should be 300 GeV. Therefore, the total kinetic energy of the two fragments is 3 GeV. But that's just the initial energy. Wait, but the fragments are moving, so their kinetic energies plus their rest energies sum up to the initial energy. So, total energy after fission is 297 GeV (rest) + kinetic energy (KE) = 300 GeV. Therefore, KE_total = 3 GeV. But since the fragments are moving, their KE is relativistic unless they are moving at non-relativistic speeds, which the classical approximation would assume.

Wait, but the problem is about the difference between the correct relativistic T1 and the classical T1. So we need to calculate T1 relativistically and then also calculate it using the classical approximation, then find the difference.

The more massive fragment is the one with mass 2m, which is 0.66M. Let's denote the more massive fragment as fragment 1 (mass 2m) and the less massive as fragment 2 (mass m). The total rest mass is 297 GeV, but the initial mass was M=300 GeV. So the total kinetic energy of the two fragments is 3 GeV. But the problem states that T1 is the kinetic energy of the more massive fragment. So in reality, the total kinetic energy is the sum of the kinetic energies of both fragments. However, the problem says "the sum of rest-masses of the two fragments is 99% of the initial mass M." So that's 297 GeV, which is 3 GeV less than M. So the total kinetic energy is 3 GeV. But this is the sum of the kinetic energies of both fragments. However, the problem states that T1 is the kinetic energy of the more massive fragment. So we need to find T1 relativistically and then using the classical approximation, then find the difference between them.

But wait, maybe I need to approach this using conservation of energy and momentum. Since the nucleus was initially at rest, the total momentum before fission is zero. After fission, the two fragments must move in opposite directions with equal and opposite momenta to conserve momentum. So their velocities are in opposite directions, say, v1 and v2, but with |p1| = |p2|. Let's denote the more massive fragment as fragment 1 (mass 2m) and the less massive as fragment 2 (mass m). Since momentum is conserved, p1 = -p2. Therefore, their momenta are equal in magnitude and opposite in direction.

Now, let's model this using relativistic equations. The total energy of the system must be conserved. The initial energy is M c² = 300 GeV (since M is given in GeV/c², and c=1 in these units). The final energy is the sum of the rest energies of the fragments plus their kinetic energies. So:

Initial energy: E_initial = M c² = 300 GeV.

Final energy: E_final = (2m + m) c² + KE1 + KE2 = 3m c² + KE1 + KE2.

But 3m c² = 0.99 M c² = 0.99 * 300 GeV = 297 GeV. Therefore, E_final = 297 GeV + KE1 + KE2 = 300 GeV. So KE1 + KE2 = 3 GeV.

But the problem states that the sum of the rest masses is 99% of M, so the rest mass of the fragments is 297 GeV, which is 0.99*300 GeV. The remaining energy is the total kinetic energy, which is 3 GeV. But the problem asks for the kinetic energy of the more massive fragment, T1. So we have to distribute this 3 GeV between the two fragments. Let's denote T1 as the kinetic energy of fragment 1 (mass 2m) and T2 as the kinetic energy of fragment 2 (mass m). Then T1 + T2 = 3 GeV.

To find T1, we need to model the relativistic expressions for the kinetic energies. The relativistic kinetic energy is given by KE = (γ - 1) m c². Since we're working in units where c=1, this simplifies to KE = (γ - 1) m. Wait, but m here is the rest mass in GeV/c². But in our units, since c=1, the rest energy is m (GeV/c²) * c² = m GeV. So the rest energy of fragment 1 is 2m * c² = 2m GeV. Wait, but earlier we had 3m = 0.99 M. Since M is 300 GeV/c², then 3m = 0.99 * 300 GeV → m = 99 GeV/c². So fragment 1 has rest mass 2m = 198 GeV/c², and fragment 2 has rest mass m = 99 GeV/c². Therefore, their rest energies are 198 GeV and 99 GeV, respectively.

So the total rest energy of the fragments is 198 + 99 = 297 GeV, as given. The total energy is 300 GeV, so the total kinetic energy is 3 GeV. Now, the kinetic energy of each fragment is (γ1 - 1)*198 GeV and (γ2 - 1)*99 GeV. But the problem is that we have two unknowns, γ1 and γ2, and two equations: one from energy conservation and one from momentum conservation.

Since the total momentum is zero before fission, the momenta of the two fragments must be equal in magnitude and opposite in direction. Therefore, the magnitude of their momenta p1 and p2 are equal: p1 = p2. But in terms of relativistic momentum, p = γ m v. However, since the velocities are relativistic, we need to express this in terms of γ. Alternatively, we can use the relation between energy and momentum for each fragment. For each fragment, the energy is given by E = γ m, and the momentum is p = γ m v. But v can be expressed in terms of γ as v = c * sqrt(1 - (1/γ²)) ≈ c (1 - 1/(2γ²)) for non-relativistic cases. However, since we're dealing with relativistic speeds, we need a better approach.

Alternatively, we can use the relation between energy and momentum for each particle. The energy of each fragment is E = sqrt(p² c² + m² c⁴). Again, since we're in units where c=1, E = sqrt(p² + m²). Therefore, for fragment 1: E1 = sqrt(p² + (198)^2), and for fragment 2: E2 = sqrt(p² + (99)^2). The total energy is E1 + E2 = 300 GeV. The total kinetic energy is (E1 - 198) + (E2 - 99) = 3 GeV. So E1 + E2 = 300 GeV, and (E1 + E2) - (198 + 99) = 3 GeV. That checks out. So we have E1 + E2 = 300 GeV, and E1 = sqrt(p² + 198²), E2 = sqrt(p² + 99²). So we can set up the equation:

sqrt(p² + 198²) + sqrt(p² + 99²) = 300.

This is one equation with one variable p. Let's solve for p.

Let me denote S = sqrt(p² + 198²) + sqrt(p² + 99²) = 300.

Let me square both sides to eliminate the square roots:

[sqrt(p² + 198²) + sqrt(p² + 99²)]² = 300²

Expanding the left side:

(p² + 198²) + (p² + 99²) + 2 sqrt{(p² + 198²)(p² + 99²)} = 90000

Simplify:

2p² + (198² + 99²) + 2 sqrt{(p² + 198²)(p² + 99²)} = 90000

Calculate 198² + 99²:

198² = (200 - 2)² = 40000 - 800 + 4 = 39204

99² = 9801

So 39204 + 9801 = 49005

Therefore:

2p² + 49005 + 2 sqrt{(p² + 39204)(p² + 9801)} = 90000

Subtract 49005 from both sides:

2p² + 2 sqrt{(p² + 39204)(p² + 9801)} = 90000 - 49005 = 40995

Divide both sides by 2:

p² + sqrt{(p² + 39204)(p² + 9801)} = 20497.5

Let me denote x = p². Then the equation becomes:

x + sqrt{(x + 39204)(x + 9801)} = 20497.5

Let me rewrite this as:

sqrt{(x + 39204)(x + 9801)} = 20497.5 - x

Now square both sides:

(x + 39204)(x + 9801) = (20497.5 - x)²

Expand both sides:

Left side: x² + (39204 + 9801)x + 39204*9801

Right side: (20497.5)^2 - 2*20497.5 x + x²

Subtract x² from both sides:

(39204 + 9801)x + 39204*9801 = (20497.5)^2 - 2*20497.5 x

Combine like terms:

(39204 + 9801 + 2*20497.5)x = (20497.5)^2 - 39204*9801

Calculate coefficients:

39204 + 9801 = 49005

2*20497.5 = 40995

Left side coefficient: 49005 + 40995 = 90000

Right side: (20497.5)^2 - 39204*9801

Calculate (20497.5)^2:

20497.5^2 = (20500 - 2.5)^2 = 20500² - 2*20500*2.5 + 2.5² = 420250000 - 102500 + 6.25 = 420147506.25

39204 * 9801: Let's compute this. 39204 * 9801. Let's note that 9801 = 99², and 39204 = 198². So 198² * 99² = (198*99)^2. 198*99 = 198*(100 -1) = 19800 - 198 = 19602. Therefore, 19602². Let's compute 19602²: 19600² + 2*19600*2 + 2² = (19600)^2 + 78400 + 4 = 384160000 + 78400 + 4 = 384238404.

Wait, but 39204 * 9801 = (198^2) * (99^2) = (198*99)^2 = (19602)^2 = 384238404. So the right side becomes:

420147506.25 - 384238404 = 35909102.25

Therefore, the equation is:

90000 x = 35909102.25

So x = 35909102.25 / 90000 ≈ 398.99 GeV²

Therefore, p² ≈ 398.99 GeV² → p ≈ sqrt(398.99) ≈ 19.975 GeV

So the momentum of each fragment is approximately 19.975 GeV/c. Let's check if this makes sense.

Now, we can find the energies of each fragment:

E1 = sqrt(p² + m1²) = sqrt(398.99 + (198)^2) = sqrt(398.99 + 39204) = sqrt(39602.99) ≈ 199 GeV

Similarly, E2 = sqrt(398.99 + 99²) = sqrt(398.99 + 9801) = sqrt(10199.99) ≈ 100.995 GeV

Check E1 + E2 ≈ 199 + 100.995 ≈ 299.995 GeV, which is approximately 300 GeV. Close enough, considering rounding.

Therefore, the kinetic energies are:

T1 = E1 - m1 = 199 - 198 = 1 GeV

T2 = E2 - m2 = 100.995 - 99 ≈ 1.995 GeV

Total kinetic energy ≈ 1 + 1.995 ≈ 3 GeV, which matches the earlier result. So in the relativistic case, T1 is approximately 1 GeV.

Now, using the classical approximation. In classical mechanics, kinetic energy is (1/2)mv², and momentum p = mv. Since the nucleus was at rest, the total momentum after fission must be zero, so the momenta of the two fragments are equal in magnitude and opposite in direction. Let's denote their velocities as v1 and v2. Then p1 = m1 v1, p2 = m2 v2. Since p1 = -p2, m1 v1 = m2 v2. Given that m1 = 2m2 (since fragment 1 is twice as massive as fragment 2?), wait wait, no. Wait, earlier, we denoted fragment 1 as the more massive one (2m), and fragment 2 as m. Wait, in the problem statement, "one fragment is 2 times more massive than the other (in terms of their rest-masses)." So if fragment 1 is 2 times more massive than fragment 2, then m1 = 2 m2. But earlier, we had m1 + m2 = 3m = 0.99M. Wait, perhaps I made a mistake earlier. Let me re-examine the problem.

Problem states: "one fragment is 2 times more massive than the other (in terms of their rest-masses)." So if fragment 1 is 2 times more massive than fragment 2, then m1 = 2 m2. The sum of their rest masses is 3m = 0.99M. So m1 + m2 = 3m = 0.99M → 2m2 + m2 = 3m2 = 0.99M → m2 = 0.33M. Then m1 = 2*0.33M = 0.66M. Therefore, fragment 1 has mass 0.66M, fragment 2 has mass 0.33M.

Wait, but M is the initial rest mass, which is 300 GeV. So m1 = 0.66*300 = 198 GeV/c², m2 = 0.33*300 = 99 GeV/c². So fragment 1 is 198 GeV, fragment 2 is 99 GeV. So fragment 1 is twice as massive as fragment 2. So in classical terms, their momenta would be p1 = m1 v1, p2 = m2 v2. Since p1 = -p2, then m1 v1 = m2 v2 → v1 = (m2/m1) v2 = (99/198) v2 = 0.5 v2. So v1 = 0.5 v2.

But in classical mechanics, kinetic energy is (1/2) m v². The total kinetic energy is T1 + T2 = (1/2) m1 v1² + (1/2) m2 v2². Since v1 = 0.5 v2, substitute:

T1 = (1/2) * 198 * (0.5 v2)^2 = (1/2)*198*(0.25 v2²) = 24.75 v2²

T2 = (1/2)*99*v2² = 49.5 v2²

Total kinetic energy: 24.75 v2² + 49.5 v2² = 74.25 v2² = 3 GeV → v2² = 3 / 74.25 ≈ 0.0404 GeV²/c² → v2 ≈ sqrt(0.0404) ≈ 0.201 GeV/c. But wait, in classical mechanics, the kinetic energy is (1/2) m v². However, if we want to find v in terms of momentum p, in classical terms, p = m v → v = p/m. Then KE = (1/2) m (p/m)^2 = p²/(2m). Since the momenta are equal in magnitude (p1 = p2 = p), then KE1 = p²/(2m1), KE2 = p²/(2m2). Therefore, total KE = p²/(2m1) + p²/(2m2) = p² (1/(2m1) + 1/(2m2)) = (p² / 2)(1/m1 + 1/m2). Since m1 = 198 GeV, m2 = 99 GeV, then 1/m1 + 1/m2 = 1/198 + 1/99 = (1 + 2)/198 = 3/198 = 1/66. Therefore, total KE = (p² / 2)(1/66) = p² / 132. The total KE is 3 GeV, so p² = 3 GeV * 132 = 396 GeV² → p = sqrt(396) ≈ 19.899 GeV/c.

Therefore, the momentum p of each fragment is approximately 19.9 GeV/c. Then, the KE1 in classical approximation is p²/(2m1) = 396 / (2*198) = 396 / 396 = 1 GeV. Similarly, KE2 = 396 / (2*99) = 396 / 198 = 2 GeV. Therefore, T1 (classical) = 1 GeV, T1 (relativistic) ≈ 1 GeV. Wait, but according to the relativistic calculation, T1 was approximately 1 GeV. But the options are 20, 10, 5, 2 MeV. Wait, that's 0.02, 0.01, 0.005, 0.002 GeV. But according to this, the difference would be zero, which isn't an option. So something must be wrong here.

Wait, perhaps I made a mistake in the classical calculation. Let's re-examine the classical approach.

In the classical approximation, the kinetic energy of each fragment is (1/2) m v². The total kinetic energy is 3 GeV. The momenta are equal in magnitude and opposite in direction. Let's denote the velocity of fragment 1 as v1 and fragment 2 as v2. Then, since m1 v1 = m2 v2 = p (magnitude). So v1 = p/m1, v2 = p/m2. Then KE1 = (1/2) m1 (p/m1)^2 = p²/(2 m1). Similarly, KE2 = p²/(2 m2). So total KE = p²/(2 m1) + p²/(2 m2) = p² (1/(2 m1) + 1/(2 m2)) = p² (1/2)(1/m1 + 1/m2). As before, 1/m1 + 1/m2 = 1/198 + 1/99 = 3/198 = 1/66. Therefore, total KE = p² * 1/132. So 3 GeV = p² / 132 → p² = 396 GeV² → p = sqrt(396) ≈ 19.899 GeV/c. Then KE1 = 396 / (2*198) = 1 GeV. KE2 = 396 / (2*99) = 2 GeV. So T1 (classical) = 1 GeV, T1 (relativistic) ≈ 1 GeV. The difference is zero. But the answer choices are in MeV, so 0.001 GeV. But the options are 20 MeV, etc. So this suggests that something is wrong with the approach.

Wait, perhaps the problem is that in the relativistic case, the kinetic energy is (gamma - 1)m. But in the relativistic case, when the mass is 198 GeV, the kinetic energy is (gamma - 1)*198 GeV. But the relativistic calculation gave T1 ≈ 1 GeV, and the classical also gives 1 GeV. But the answer options are much smaller. So perhaps I made a mistake in the problem setup.

Wait, let's re-examine the problem statement. The sum of rest-masses is 99% of the initial mass M. So sum of rest masses is 0.99 M = 297 GeV. But M is 300 GeV. Then the total kinetic energy is 3 GeV. But 3 GeV is 3000 MeV. The answer options are in MeV. So 3 GeV is 3000 MeV. But the options are 20, 10, 5, 2 MeV. So perhaps there is an error in the calculation.

Wait, perhaps the mistake is that in the relativistic case, the total kinetic energy is not 3 GeV. Wait, the initial rest energy is 300 GeV. The sum of rest energies after fission is 297 GeV. Therefore, the total energy after fission is 300 GeV (since energy is conserved in the rest frame). Therefore, the total kinetic energy is 300 - 297 = 3 GeV. So that part is correct. Therefore, the total kinetic energy is 3 GeV, which is 3000 MeV.

But according to the relativistic calculation, the kinetic energy of fragment 1 is approximately 1 GeV (1000 MeV), and classical gives 1 GeV as well. But the problem says to find the difference between the correct T1 and the classical approximation. But according to this, the difference is zero, which is not an option. Therefore, I must have made a mistake in the setup.

Wait, perhaps the problem is that in the relativistic calculation, the kinetic energy is (gamma -1) m, but for the rest mass of fragment 1 being 198 GeV. Let's recalculate the relativistic kinetic energy more accurately.

Earlier, we found that p ≈ 19.975 GeV/c. Then, using the relativistic formula for kinetic energy: KE = (gamma - 1) m, where gamma = 1 / sqrt(1 - (v²/c²)).

But since p = gamma m v, we can express v in terms of p and m.

v = p / (gamma m). Then, gamma = 1 / sqrt(1 - (v²/c²)) = 1 / sqrt(1 - (p²)/(gamma² m²)).

This seems a bit circular. Alternatively, using the relation between energy and momentum: E = sqrt(p² + m²). Therefore, gamma = E / m.

So gamma = sqrt(p² + m²) / m = sqrt(1 + (p²)/(m²)).

Therefore, KE = (gamma - 1) m = [sqrt(1 + (p²)/(m²)) - 1] m.

Let's compute this for fragment 1: m1 = 198 GeV/c², p = 19.975 GeV/c.

p² = (19.975)^2 ≈ 399 GeV².

So (p²)/(m1²) = 399 / (198)^2 ≈ 399 / 39204 ≈ 0.01014.

Therefore, sqrt(1 + 0.01014) ≈ sqrt(1.01014) ≈ 1.00505.

Thus, gamma ≈ 1.00505.

Therefore, KE1 = (gamma - 1) * m1 ≈ (0.00505) * 198 ≈ 0.00505 * 200 ≈ 1.01 GeV. But more accurately, 0.00505 * 198 = 1.0009 GeV.

Similarly, for fragment 2: m2 = 99 GeV/c², p = 19.975 GeV/c.

(p²)/(m2²) = 399 / (99)^2 ≈ 399 / 9801 ≈ 0.04075.

sqrt(1 + 0.04075) ≈ sqrt(1.04075) ≈ 1.0201.

gamma ≈ 1.0201.

KE2 = (gamma - 1) * m2 ≈ 0.0201 * 99 ≈ 1.99 GeV.

So total kinetic energy is 1.01 + 1.99 ≈ 3 GeV, which matches. Therefore, KE1 (relativistic) is approximately 1.01 GeV, KE2 ≈ 1.99 GeV.

But the classical calculation gave KE1 = 1 GeV, KE2 = 2 GeV. So the difference between the relativistic and classical KE1 is 1.01 GeV - 1 GeV = 0.01 GeV = 10 MeV. But the options are 20, 10, 5, 2 MeV. So 10 MeV is option B. However, the problem asks for the difference between the correct T1 (relativistic) and T1 (classical), which would be 10 MeV. But according to the problem statement, the answer options are A.20, B.10, etc.

But wait, the problem states that the sum of the rest masses is 99% of the initial mass. So 297 GeV. Then total kinetic energy is 3 GeV. But according to the relativistic calculation, the kinetic energy of fragment 1 is approximately 1.01 GeV (1010 MeV), and classically it's 1000 MeV. So difference is 10 MeV, which is option B. But the initial calculation suggested that both relativistic and classical gave 1 GeV, but that was a mistake because I forgot that in the relativistic case, the gamma factor gives a slightly higher energy.

Wait, let me verify again. The relativistic KE1 is approximately 1.01 GeV, and classical is 1 GeV. The difference is 0.01 GeV = 10 MeV. So the answer should be B.10 MeV. But the options given are:

A. 20 MeV.

B. 10 MeV.

C. 5 MeV.

D. 2 MeV.

So according to this, the answer is B.10 MeV. But let's check if there's any mistake in the setup.

Wait, perhaps I made a mistake in the problem setup. Let's re-examine the problem statement.

The problem states that the sum of the rest-masses of the two fragments is 99% of the initial mass M. So 0.99 M. But the initial mass M is 300 GeV. So 0.99 * 300 = 297 GeV. Therefore, the total rest energy after fission is 297 GeV, and the total energy is 300 GeV. Therefore, the total kinetic energy is 3 GeV. So that's correct.

But according to the relativistic calculation, the kinetic energy of fragment 1 is approximately 1.01 GeV, and classically it's 1 GeV. So the difference is 0.01 GeV = 10 MeV. Therefore, the answer should be B.10 MeV.

But wait, the problem says "the difference between the (correct) T1 value, and T1 value calculated using classical (non-relativistic) approximation?" So correct T1 (relativistic) is 1.01 GeV, classical is 1 GeV. Difference is 0.01 GeV = 10 MeV. So answer is B.10 MeV.

But let me check once more. The relativistic KE1 is (gamma - 1) * m1. Gamma for fragment 1 is sqrt(1 + (p²)/(m1²)).

p = sqrt(396) ≈ 19.975 GeV/c.

p² = 396.

m1² = 198² = 39204.

(p²)/(m1²) = 396/39204 ≈ 0.0101.

sqrt(1 + 0.0101) ≈ 1.005037.

gamma ≈ 1.005037.

KE1 = (1.005037 - 1) * 198 ≈ 0.005037 * 198 ≈ 1.000 GeV. Wait, that's approximately 1 GeV. Wait, but earlier calculation with more precise gamma gave 1.01 GeV. Wait, perhaps my approximation was off. Let's compute gamma more accurately.

gamma = sqrt(1 + 0.0101) = sqrt(1.0101) ≈ 1.005037.

So KE1 = (1.005037 - 1) * 198 = 0.005037 * 198 ≈ 1.000 GeV. So actually, the relativistic KE1 is approximately 1 GeV, same as classical. But this contradicts the earlier precise calculation. Wait, perhaps my earlier calculation was incorrect.

Wait, let's compute gamma more accurately. Let's compute sqrt(1.0101):

Using Taylor expansion: sqrt(1 + x) ≈ 1 + x/2 - x²/8 + ... for small x.

Here, x = 0.0101.

sqrt(1.0101) ≈ 1 + 0.0101/2 - (0.0101)^2 /8 ≈ 1 + 0.00505 - 0.00001275 ≈ 1.005037.

Therefore, gamma ≈ 1.005037.

Therefore, KE1 = (1.005037 - 1) * 198 = 0.005037 * 198 = 1.000 GeV.

Wait, so relativistic KE1 is approximately 1 GeV, same as classical. Then the difference would be zero, which is not an option. But this contradicts the earlier calculation where using the exact formula, KE1 was 1.01 GeV. So which is correct?

Wait, perhaps the confusion comes from the fact that in the relativistic case, the total kinetic energy is 3 GeV, but the individual kinetic energies are not exactly 1 GeV each. Because in the relativistic case, the two fragments have different velocities (since their masses are different), so their kinetic energies are not equal. Wait, but earlier, we found that KE1 ≈1.01 GeV and KE2 ≈1.99 GeV, totaling 3 GeV. So the difference between relativistic and classical is fragment 1's KE is 1.01 vs 1, difference 0.01 GeV, which is 10 MeV.

But according to the more precise calculation, the kinetic energies are 1.01 and 1.99 GeV. So the difference is 0.01 GeV = 10 MeV. Therefore, the answer is B.10 MeV.

But why did the approximation using gamma -1 give 1 GeV when the exact calculation gave 1.01 GeV? Because the approximation gamma ≈1 + (p²)/(2 m²) is only valid for low velocities (non-relativistic). Wait, but in this case, the momentum p is 19.975 GeV/c, and the mass is 198 GeV/c². So the velocity v is given by v = p/(gamma m). But gamma is 1.005, so v ≈ (19.975)/(1.005 * 198) ≈ 19.975/(199) ≈ 0.1003c. So v is approximately 0.1c, which is not extremely relativistic. Therefore, the approximation gamma ≈1 + (p²)/(2 m²) should be reasonably accurate.

Using the non-relativistic formula for KE: KE = (1/2) m v².

But p = gamma m v, so v = p/(gamma m). Therefore, KE = (1/2) m (p²)/(gamma² m²) = p²/(2 gamma² m).

But since gamma is close to 1, this is approximately p²/(2 m²). However, in reality, gamma is 1.005, so the non-relativistic KE would be p²/(2 m²) * (1/gamma²) ≈ p²/(2 m²) * (1 - 2*(0.005)) ≈ p²/(2 m²) - p²/(2 m²) * 0.01.

But p²/(2 m²) is the non-relativistic KE. For fragment 1:

p²/(2 m1²) = 396/(2*39204) = 396/78408 ≈ 0.00505 GeV. Wait, that's way too small. Wait, there's a miscalculation here. Let's clarify.

Wait, the non-relativistic KE is (1/2) m v². But p = gamma m v → v = p/(gamma m). So KE = (1/2) m (p²)/(gamma² m²) = p²/(2 gamma² m). Since gamma is 1.005, gamma² ≈1.010025. So KE ≈ p²/(2*1.010025*m) ≈ p²/(2 m) * (1 - 0.010025). So the non-relativistic KE would be approximately p²/(2 m) * 0.99. Therefore, the relativistic KE is slightly higher than the non-relativistic KE by a factor of 1/gamma² ≈1.010025. Therefore, the relativistic KE is approximately 1.01 times the non-relativistic KE. Therefore, if the non-relativistic KE is 1 GeV, the relativistic KE would be approximately 1.01 GeV, which matches the exact calculation.

But according to the exact calculation, the non-relativistic KE is 1 GeV, and relativistic is approximately 1.01 GeV. Therefore, the difference is 0.01 GeV = 10 MeV.

Therefore, the correct answer is B.10 MeV.
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