normal shock relationships in a supersonic wind tunnel diffuser Given upstream Mach number (M_1) = 2.4000; specific heat ratio (k) = 1.3600, determine the downstream Mach number (M_2).

Start with the thinking

• The governing relation printed in this handbook section is Normal shock relationships.
• Everything except M_2 is given, so isolate M_2 symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Normal shock relationships give the downstream Mach number for supersonic flow crossing a normal shock.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$M_2^2 = \dfrac{M_1^2 + \dfrac{2}{k-1}}{\dfrac{2k}{k-1}M_1^2 - 1}$$

2. Step 2 — Rearrange the relation so that M_2 stands alone on the left-hand side.

3. Step 3 — List the givens: upstream Mach number (M_1) = 2.4000, specific heat ratio (k) = 1.3600.

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$M_{2} = 0.5159$$

6. Step 6 — Check: returning M_2 = 0.5159 to

   $$M_2^2 = \dfrac{M_1^2 + \dfrac{2}{k-1}}{\dfrac{2k}{k-1}M_1^2 - 1}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 1.0317 — kept a factor of two that cancels in the correct rearrangement.
• 0.2579 — dropped that same factor in the other direction.
• 0.5675 — rounded an intermediate value before the final step.

Reference: FE Handbook — Normal Shock Relationships

normal shock relationships ahead of a blunt body in supersonic flow Given upstream Mach number (M_1) = 1.2500; specific heat ratio (k) = 1.3800, determine the downstream Mach number (M_2).

Start with the thinking

• The governing relation printed in this handbook section is Normal shock relationships.
• Everything except M_2 is given, so isolate M_2 symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Normal shock relationships give the downstream Mach number for supersonic flow crossing a normal shock.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$M_2^2 = \dfrac{M_1^2 + \dfrac{2}{k-1}}{\dfrac{2k}{k-1}M_1^2 - 1}$$

2. Step 2 — Rearrange the relation so that M_2 stands alone on the left-hand side.

3. Step 3 — List the givens: upstream Mach number (M_1) = 1.2500, specific heat ratio (k) = 1.3800.

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$M_{2} = 0.8121$$

6. Step 6 — Check: returning M_2 = 0.8121 to

   $$M_2^2 = \dfrac{M_1^2 + \dfrac{2}{k-1}}{\dfrac{2k}{k-1}M_1^2 - 1}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 1.6243 — kept a factor of two that cancels in the correct rearrangement.
• 0.4061 — dropped that same factor in the other direction.
• 0.8934 — rounded an intermediate value before the final step.

Reference: FE Handbook — Normal Shock Relationships

normal shock relationships for flow in a supersonic nozzle Given upstream Mach number (M_1) = 1.2000; specific heat ratio (k) = 1.3500, determine the downstream Mach number (M_2).

Start with the thinking

• The governing relation printed in this handbook section is Normal shock relationships.
• Everything except M_2 is given, so isolate M_2 symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Normal shock relationships give the downstream Mach number for supersonic flow crossing a normal shock.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$M_2^2 = \dfrac{M_1^2 + \dfrac{2}{k-1}}{\dfrac{2k}{k-1}M_1^2 - 1}$$

2. Step 2 — Rearrange the relation so that M_2 stands alone on the left-hand side.

3. Step 3 — List the givens: upstream Mach number (M_1) = 1.2000, specific heat ratio (k) = 1.3500.

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$M_{2} = 0.8413$$

6. Step 6 — Check: returning M_2 = 0.8413 to

   $$M_2^2 = \dfrac{M_1^2 + \dfrac{2}{k-1}}{\dfrac{2k}{k-1}M_1^2 - 1}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 1.6826 — kept a factor of two that cancels in the correct rearrangement.
• 0.4206 — dropped that same factor in the other direction.
• 0.9254 — rounded an intermediate value before the final step.

Reference: FE Handbook — Normal Shock Relationships

normal shock relationships in a supersonic wind tunnel diffuser Given upstream Mach number (M_1) = 2.4500; specific heat ratio (k) = 1.3900, determine the downstream Mach number (M_2).

Start with the thinking

• The governing relation printed in this handbook section is Normal shock relationships.
• Everything except M_2 is given, so isolate M_2 symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Normal shock relationships give the downstream Mach number for supersonic flow crossing a normal shock.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$M_2^2 = \dfrac{M_1^2 + \dfrac{2}{k-1}}{\dfrac{2k}{k-1}M_1^2 - 1}$$

2. Step 2 — Rearrange the relation so that M_2 stands alone on the left-hand side.

3. Step 3 — List the givens: upstream Mach number (M_1) = 2.4500, specific heat ratio (k) = 1.3900.

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$M_{2} = 0.5161$$

6. Step 6 — Check: returning M_2 = 0.5161 to

   $$M_2^2 = \dfrac{M_1^2 + \dfrac{2}{k-1}}{\dfrac{2k}{k-1}M_1^2 - 1}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 1.0322 — kept a factor of two that cancels in the correct rearrangement.
• 0.2581 — dropped that same factor in the other direction.
• 0.5677 — rounded an intermediate value before the final step.

Reference: FE Handbook — Normal Shock Relationships

normal shock relationships ahead of a blunt body in supersonic flow Given upstream Mach number (M_1) = 1.7500; specific heat ratio (k) = 1.3500, determine the downstream Mach number (M_2).

Start with the thinking

• The governing relation printed in this handbook section is Normal shock relationships.
• Everything except M_2 is given, so isolate M_2 symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Normal shock relationships give the downstream Mach number for supersonic flow crossing a normal shock.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$M_2^2 = \dfrac{M_1^2 + \dfrac{2}{k-1}}{\dfrac{2k}{k-1}M_1^2 - 1}$$

2. Step 2 — Rearrange the relation so that M_2 stands alone on the left-hand side.

3. Step 3 — List the givens: upstream Mach number (M_1) = 1.7500, specific heat ratio (k) = 1.3500.

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$M_{2} = 0.6228$$

6. Step 6 — Check: returning M_2 = 0.6228 to

   $$M_2^2 = \dfrac{M_1^2 + \dfrac{2}{k-1}}{\dfrac{2k}{k-1}M_1^2 - 1}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 1.2457 — kept a factor of two that cancels in the correct rearrangement.
• 0.3114 — dropped that same factor in the other direction.
• 0.6851 — rounded an intermediate value before the final step.

Reference: FE Handbook — Normal Shock Relationships

normal shock relationships for flow in a supersonic nozzle Given upstream Mach number (M_1) = 1.3000; specific heat ratio (k) = 1.3600, determine the downstream Mach number (M_2).

Start with the thinking

• The governing relation printed in this handbook section is Normal shock relationships.
• Everything except M_2 is given, so isolate M_2 symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Normal shock relationships give the downstream Mach number for supersonic flow crossing a normal shock.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$M_2^2 = \dfrac{M_1^2 + \dfrac{2}{k-1}}{\dfrac{2k}{k-1}M_1^2 - 1}$$

2. Step 2 — Rearrange the relation so that M_2 stands alone on the left-hand side.

3. Step 3 — List the givens: upstream Mach number (M_1) = 1.3000, specific heat ratio (k) = 1.3600.

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$M_{2} = 0.7846$$

6. Step 6 — Check: returning M_2 = 0.7846 to

   $$M_2^2 = \dfrac{M_1^2 + \dfrac{2}{k-1}}{\dfrac{2k}{k-1}M_1^2 - 1}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 1.5693 — kept a factor of two that cancels in the correct rearrangement.
• 0.3923 — dropped that same factor in the other direction.
• 0.8631 — rounded an intermediate value before the final step.

Reference: FE Handbook — Normal Shock Relationships

normal shock relationships in a supersonic wind tunnel diffuser Given upstream Mach number (M_1) = 2.5500; specific heat ratio (k) = 1.3900, determine the downstream Mach number (M_2).

Start with the thinking

• The governing relation printed in this handbook section is Normal shock relationships.
• Everything except M_2 is given, so isolate M_2 symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Normal shock relationships give the downstream Mach number for supersonic flow crossing a normal shock.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$M_2^2 = \dfrac{M_1^2 + \dfrac{2}{k-1}}{\dfrac{2k}{k-1}M_1^2 - 1}$$

2. Step 2 — Rearrange the relation so that M_2 stands alone on the left-hand side.

3. Step 3 — List the givens: upstream Mach number (M_1) = 2.5500, specific heat ratio (k) = 1.3900.

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$M_{2} = 0.5064$$

6. Step 6 — Check: returning M_2 = 0.5064 to

   $$M_2^2 = \dfrac{M_1^2 + \dfrac{2}{k-1}}{\dfrac{2k}{k-1}M_1^2 - 1}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 1.0128 — kept a factor of two that cancels in the correct rearrangement.
• 0.2532 — dropped that same factor in the other direction.
• 0.5571 — rounded an intermediate value before the final step.

Reference: FE Handbook — Normal Shock Relationships

normal shock relationships ahead of a blunt body in supersonic flow Given upstream Mach number (M_1) = 1.4000; specific heat ratio (k) = 1.3900, determine the downstream Mach number (M_2).

Start with the thinking

• The governing relation printed in this handbook section is Normal shock relationships.
• Everything except M_2 is given, so isolate M_2 symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Normal shock relationships give the downstream Mach number for supersonic flow crossing a normal shock.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$M_2^2 = \dfrac{M_1^2 + \dfrac{2}{k-1}}{\dfrac{2k}{k-1}M_1^2 - 1}$$

2. Step 2 — Rearrange the relation so that M_2 stands alone on the left-hand side.

3. Step 3 — List the givens: upstream Mach number (M_1) = 1.4000, specific heat ratio (k) = 1.3900.

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$M_{2} = 0.7392$$

6. Step 6 — Check: returning M_2 = 0.7392 to

   $$M_2^2 = \dfrac{M_1^2 + \dfrac{2}{k-1}}{\dfrac{2k}{k-1}M_1^2 - 1}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 1.4785 — kept a factor of two that cancels in the correct rearrangement.
• 0.3696 — dropped that same factor in the other direction.
• 0.8131 — rounded an intermediate value before the final step.

Reference: FE Handbook — Normal Shock Relationships

normal shock relationships for flow in a supersonic nozzle Given upstream Mach number (M_1) = 2.9500; specific heat ratio (k) = 1.3400, determine the downstream Mach number (M_2).

Start with the thinking

• The governing relation printed in this handbook section is Normal shock relationships.
• Everything except M_2 is given, so isolate M_2 symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Normal shock relationships give the downstream Mach number for supersonic flow crossing a normal shock.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$M_2^2 = \dfrac{M_1^2 + \dfrac{2}{k-1}}{\dfrac{2k}{k-1}M_1^2 - 1}$$

2. Step 2 — Rearrange the relation so that M_2 stands alone on the left-hand side.

3. Step 3 — List the givens: upstream Mach number (M_1) = 2.9500, specific heat ratio (k) = 1.3400.

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$M_{2} = 0.4645$$

6. Step 6 — Check: returning M_2 = 0.4645 to

   $$M_2^2 = \dfrac{M_1^2 + \dfrac{2}{k-1}}{\dfrac{2k}{k-1}M_1^2 - 1}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 0.9290 — kept a factor of two that cancels in the correct rearrangement.
• 0.2323 — dropped that same factor in the other direction.
• 0.5110 — rounded an intermediate value before the final step.

Reference: FE Handbook — Normal Shock Relationships

normal shock relationships in a supersonic wind tunnel diffuser Given upstream Mach number (M_1) = 2.1500; specific heat ratio (k) = 1.3700, determine the downstream Mach number (M_2).

Start with the thinking

• The governing relation printed in this handbook section is Normal shock relationships.
• Everything except M_2 is given, so isolate M_2 symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Normal shock relationships give the downstream Mach number for supersonic flow crossing a normal shock.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$M_2^2 = \dfrac{M_1^2 + \dfrac{2}{k-1}}{\dfrac{2k}{k-1}M_1^2 - 1}$$

2. Step 2 — Rearrange the relation so that M_2 stands alone on the left-hand side.

3. Step 3 — List the givens: upstream Mach number (M_1) = 2.1500, specific heat ratio (k) = 1.3700.

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$M_{2} = 0.5493$$

6. Step 6 — Check: returning M_2 = 0.5493 to

   $$M_2^2 = \dfrac{M_1^2 + \dfrac{2}{k-1}}{\dfrac{2k}{k-1}M_1^2 - 1}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 1.0987 — kept a factor of two that cancels in the correct rearrangement.
• 0.2747 — dropped that same factor in the other direction.
• 0.6043 — rounded an intermediate value before the final step.

Reference: FE Handbook — Normal Shock Relationships