Two billiard balls undergo a direct central impact on a table. Given velocity 1 before impact (v_1) = 15.5000 ft/s; velocity 2 before impact (v_2) = 15.0000 ft/s; velocity 1 after impact (v_1p) = 8.0000 ft/s; velocity 2 after impact (v_2p) = 14.5000 ft/s, determine the coefficient of restitution (e).

Start with the thinking

• The governing relation printed in this handbook section is Coefficient of restitution (direct central impact).
• Everything except e is given, so isolate e 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.
• Direct central impact between two bodies uses the coefficient of restitution to relate velocities before and after impact.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$e = \dfrac{v_2' - v_1'}{v_1 - v_2}$$

2. Step 2 — Rearrange symbolically for e:

   $$e = \dfrac{v_2' - v_1'}{v_1 - v_2}$$

3. Step 3 — List the givens: velocity 1 before impact (v_1) = 15.5000 ft/s, velocity 2 before impact (v_2) = 15.0000 ft/s, velocity 1 after impact (v_1p) = 8.0000 ft/s, velocity 2 after impact (v_2p) = 14.5000 ft/s.

4. Step 4 — Substitute the given values:

   $$e = \dfrac{v_2' - v_1'}{v_1 - v_2}$$

5. Step 5 — Evaluate:

   $$e = 13.0000$$

6. Step 6 — Check: returning e = 13.0000 to

   $$e = \dfrac{v_2' - v_1'}{v_1 - v_2}$$

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

Why the other options are there

• 26.0000 — kept a factor of two that cancels in the correct rearrangement.
• 6.5000 — dropped that same factor in the other direction.
• 14.3000 — rounded an intermediate value before the final step.

Reference: FE Handbook — Dynamics: Impact

A hammer strikes a pile in an impact analysis for driving energy. Given velocity 1 before impact (v_1) = 7.5000 ft/s; velocity 2 before impact (v_2) = 14.0000 ft/s; velocity 1 after impact (v_1p) = 3.0000 ft/s; coefficient of restitution (e) = 0.6400, determine the velocity 2 after impact (v_2p) in ft/s.

Start with the thinking

• The governing relation printed in this handbook section is Coefficient of restitution (direct central impact).
• Everything except v_2p is given, so isolate v_2p 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.
• Direct central impact between two bodies uses the coefficient of restitution to relate velocities before and after impact.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$e = \dfrac{v_2' - v_1'}{v_1 - v_2}$$

2. Step 2 — Rearrange symbolically for v_2p:

   $$v_{2p} = e (v_1 - v_2) + v_1'$$

3. Step 3 — List the givens: velocity 1 before impact (v_1) = 7.5000 ft/s, velocity 2 before impact (v_2) = 14.0000 ft/s, velocity 1 after impact (v_1p) = 3.0000 ft/s, coefficient of restitution (e) = 0.6400.

4. Step 4 — Substitute the given values:

   $$v_{2p} = 0.6400 (v_1 - v_2) + v_1'$$

5. Step 5 — Evaluate:

   $$v_{2p} = -1.1600\ \text{ft/s}$$

6. Step 6 — Check: returning v_2p = -1.1600 ft/s to

   $$e = \dfrac{v_2' - v_1'}{v_1 - v_2}$$

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

Why the other options are there

• -2.3200 — kept a factor of two that cancels in the correct rearrangement.
• -0.5800 — dropped that same factor in the other direction.
• -1.2760 — rounded an intermediate value before the final step.

Reference: FE Handbook — Dynamics: Impact

Two rail cars collide and the impact coefficient of restitution is measured. Given velocity 1 before impact (v_1) = 6.0000 ft/s; velocity 2 before impact (v_2) = 0.5000 ft/s; velocity 2 after impact (v_2p) = 30.5000 ft/s; coefficient of restitution (e) = 0.1100, determine the velocity 1 after impact (v_1p) in ft/s.

Start with the thinking

• The governing relation printed in this handbook section is Coefficient of restitution (direct central impact).
• Everything except v_1p is given, so isolate v_1p 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.
• Direct central impact between two bodies uses the coefficient of restitution to relate velocities before and after impact.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$e = \dfrac{v_2' - v_1'}{v_1 - v_2}$$

2. Step 2 — Rearrange symbolically for v_1p:

   $$v_{1p} = v_2' - e(v_1 - v_2)$$

3. Step 3 — List the givens: velocity 1 before impact (v_1) = 6.0000 ft/s, velocity 2 before impact (v_2) = 0.5000 ft/s, velocity 2 after impact (v_2p) = 30.5000 ft/s, coefficient of restitution (e) = 0.1100.

4. Step 4 — Substitute the given values:

   $$v_{1p} = v_2' - 0.1100(v_1 - v_2)$$

5. Step 5 — Evaluate:

   $$v_{1p} = 29.8950\ \text{ft/s}$$

6. Step 6 — Check: returning v_1p = 29.8950 ft/s to

   $$e = \dfrac{v_2' - v_1'}{v_1 - v_2}$$

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

Why the other options are there

• 59.7900 — kept a factor of two that cancels in the correct rearrangement.
• 14.9475 — dropped that same factor in the other direction.
• 32.8845 — rounded an intermediate value before the final step.

Reference: FE Handbook — Dynamics: Impact

Two billiard balls undergo a direct central impact on a table. Given velocity 1 before impact (v_1) = 33.5000 ft/s; velocity 2 before impact (v_2) = 12.0000 ft/s; velocity 1 after impact (v_1p) = 11.5000 ft/s; velocity 2 after impact (v_2p) = 14.5000 ft/s, determine the coefficient of restitution (e).

Start with the thinking

• The governing relation printed in this handbook section is Coefficient of restitution (direct central impact).
• Everything except e is given, so isolate e 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.
• Direct central impact between two bodies uses the coefficient of restitution to relate velocities before and after impact.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$e = \dfrac{v_2' - v_1'}{v_1 - v_2}$$

2. Step 2 — Rearrange symbolically for e:

   $$e = \dfrac{v_2' - v_1'}{v_1 - v_2}$$

3. Step 3 — List the givens: velocity 1 before impact (v_1) = 33.5000 ft/s, velocity 2 before impact (v_2) = 12.0000 ft/s, velocity 1 after impact (v_1p) = 11.5000 ft/s, velocity 2 after impact (v_2p) = 14.5000 ft/s.

4. Step 4 — Substitute the given values:

   $$e = \dfrac{v_2' - v_1'}{v_1 - v_2}$$

5. Step 5 — Evaluate:

   $$e = 0.1395$$

6. Step 6 — Check: returning e = 0.1395 to

   $$e = \dfrac{v_2' - v_1'}{v_1 - v_2}$$

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

Why the other options are there

• 0.2791 — kept a factor of two that cancels in the correct rearrangement.
• 0.0698 — dropped that same factor in the other direction.
• 0.1535 — rounded an intermediate value before the final step.

Reference: FE Handbook — Dynamics: Impact

A hammer strikes a pile in an impact analysis for driving energy. Given velocity 1 before impact (v_1) = 30.5000 ft/s; velocity 2 before impact (v_2) = 1.0000 ft/s; velocity 1 after impact (v_1p) = 7.0000 ft/s; coefficient of restitution (e) = 0.2700, determine the velocity 2 after impact (v_2p) in ft/s.

Start with the thinking

• The governing relation printed in this handbook section is Coefficient of restitution (direct central impact).
• Everything except v_2p is given, so isolate v_2p 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.
• Direct central impact between two bodies uses the coefficient of restitution to relate velocities before and after impact.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$e = \dfrac{v_2' - v_1'}{v_1 - v_2}$$

2. Step 2 — Rearrange symbolically for v_2p:

   $$v_{2p} = e (v_1 - v_2) + v_1'$$

3. Step 3 — List the givens: velocity 1 before impact (v_1) = 30.5000 ft/s, velocity 2 before impact (v_2) = 1.0000 ft/s, velocity 1 after impact (v_1p) = 7.0000 ft/s, coefficient of restitution (e) = 0.2700.

4. Step 4 — Substitute the given values:

   $$v_{2p} = 0.2700 (v_1 - v_2) + v_1'$$

5. Step 5 — Evaluate:

   $$v_{2p} = 14.9650\ \text{ft/s}$$

6. Step 6 — Check: returning v_2p = 14.9650 ft/s to

   $$e = \dfrac{v_2' - v_1'}{v_1 - v_2}$$

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

Why the other options are there

• 29.9300 — kept a factor of two that cancels in the correct rearrangement.
• 7.4825 — dropped that same factor in the other direction.
• 16.4615 — rounded an intermediate value before the final step.

Reference: FE Handbook — Dynamics: Impact

Two rail cars collide and the impact coefficient of restitution is measured. Given velocity 1 before impact (v_1) = 21.5000 ft/s; velocity 2 before impact (v_2) = 7.0000 ft/s; velocity 2 after impact (v_2p) = 10.0000 ft/s; coefficient of restitution (e) = 0.7300, determine the velocity 1 after impact (v_1p) in ft/s.

Start with the thinking

• The governing relation printed in this handbook section is Coefficient of restitution (direct central impact).
• Everything except v_1p is given, so isolate v_1p 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.
• Direct central impact between two bodies uses the coefficient of restitution to relate velocities before and after impact.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$e = \dfrac{v_2' - v_1'}{v_1 - v_2}$$

2. Step 2 — Rearrange symbolically for v_1p:

   $$v_{1p} = v_2' - e(v_1 - v_2)$$

3. Step 3 — List the givens: velocity 1 before impact (v_1) = 21.5000 ft/s, velocity 2 before impact (v_2) = 7.0000 ft/s, velocity 2 after impact (v_2p) = 10.0000 ft/s, coefficient of restitution (e) = 0.7300.

4. Step 4 — Substitute the given values:

   $$v_{1p} = v_2' - 0.7300(v_1 - v_2)$$

5. Step 5 — Evaluate:

   $$v_{1p} = -0.5850\ \text{ft/s}$$

6. Step 6 — Check: returning v_1p = -0.5850 ft/s to

   $$e = \dfrac{v_2' - v_1'}{v_1 - v_2}$$

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

Why the other options are there

• -1.1700 — kept a factor of two that cancels in the correct rearrangement.
• -0.2925 — dropped that same factor in the other direction.
• -0.6435 — rounded an intermediate value before the final step.

Reference: FE Handbook — Dynamics: Impact

Two billiard balls undergo a direct central impact on a table. Given velocity 1 before impact (v_1) = 19.0000 ft/s; velocity 2 before impact (v_2) = 13.5000 ft/s; velocity 1 after impact (v_1p) = 0.5000 ft/s; velocity 2 after impact (v_2p) = 8.5000 ft/s, determine the coefficient of restitution (e).

Start with the thinking

• The governing relation printed in this handbook section is Coefficient of restitution (direct central impact).
• Everything except e is given, so isolate e 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.
• Direct central impact between two bodies uses the coefficient of restitution to relate velocities before and after impact.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$e = \dfrac{v_2' - v_1'}{v_1 - v_2}$$

2. Step 2 — Rearrange symbolically for e:

   $$e = \dfrac{v_2' - v_1'}{v_1 - v_2}$$

3. Step 3 — List the givens: velocity 1 before impact (v_1) = 19.0000 ft/s, velocity 2 before impact (v_2) = 13.5000 ft/s, velocity 1 after impact (v_1p) = 0.5000 ft/s, velocity 2 after impact (v_2p) = 8.5000 ft/s.

4. Step 4 — Substitute the given values:

   $$e = \dfrac{v_2' - v_1'}{v_1 - v_2}$$

5. Step 5 — Evaluate:

   $$e = 1.4545$$

6. Step 6 — Check: returning e = 1.4545 to

   $$e = \dfrac{v_2' - v_1'}{v_1 - v_2}$$

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

Why the other options are there

• 2.9091 — kept a factor of two that cancels in the correct rearrangement.
• 0.7273 — dropped that same factor in the other direction.
• 1.6000 — rounded an intermediate value before the final step.

Reference: FE Handbook — Dynamics: Impact

A hammer strikes a pile in an impact analysis for driving energy. Given velocity 1 before impact (v_1) = 35.5000 ft/s; velocity 2 before impact (v_2) = 7.0000 ft/s; velocity 1 after impact (v_1p) = 6.5000 ft/s; coefficient of restitution (e) = 0.9500, determine the velocity 2 after impact (v_2p) in ft/s.

Start with the thinking

• The governing relation printed in this handbook section is Coefficient of restitution (direct central impact).
• Everything except v_2p is given, so isolate v_2p 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.
• Direct central impact between two bodies uses the coefficient of restitution to relate velocities before and after impact.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$e = \dfrac{v_2' - v_1'}{v_1 - v_2}$$

2. Step 2 — Rearrange symbolically for v_2p:

   $$v_{2p} = e (v_1 - v_2) + v_1'$$

3. Step 3 — List the givens: velocity 1 before impact (v_1) = 35.5000 ft/s, velocity 2 before impact (v_2) = 7.0000 ft/s, velocity 1 after impact (v_1p) = 6.5000 ft/s, coefficient of restitution (e) = 0.9500.

4. Step 4 — Substitute the given values:

   $$v_{2p} = 0.9500 (v_1 - v_2) + v_1'$$

5. Step 5 — Evaluate:

   $$v_{2p} = 33.5750\ \text{ft/s}$$

6. Step 6 — Check: returning v_2p = 33.5750 ft/s to

   $$e = \dfrac{v_2' - v_1'}{v_1 - v_2}$$

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

Why the other options are there

• 67.1500 — kept a factor of two that cancels in the correct rearrangement.
• 16.7875 — dropped that same factor in the other direction.
• 36.9325 — rounded an intermediate value before the final step.

Reference: FE Handbook — Dynamics: Impact

Two rail cars collide and the impact coefficient of restitution is measured. Given velocity 1 before impact (v_1) = 23.0000 ft/s; velocity 2 before impact (v_2) = 1.0000 ft/s; velocity 2 after impact (v_2p) = 8.0000 ft/s; coefficient of restitution (e) = 0.0500, determine the velocity 1 after impact (v_1p) in ft/s.

Start with the thinking

• The governing relation printed in this handbook section is Coefficient of restitution (direct central impact).
• Everything except v_1p is given, so isolate v_1p 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.
• Direct central impact between two bodies uses the coefficient of restitution to relate velocities before and after impact.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$e = \dfrac{v_2' - v_1'}{v_1 - v_2}$$

2. Step 2 — Rearrange symbolically for v_1p:

   $$v_{1p} = v_2' - e(v_1 - v_2)$$

3. Step 3 — List the givens: velocity 1 before impact (v_1) = 23.0000 ft/s, velocity 2 before impact (v_2) = 1.0000 ft/s, velocity 2 after impact (v_2p) = 8.0000 ft/s, coefficient of restitution (e) = 0.0500.

4. Step 4 — Substitute the given values:

   $$v_{1p} = v_2' - 0.0500(v_1 - v_2)$$

5. Step 5 — Evaluate:

   $$v_{1p} = 6.9000\ \text{ft/s}$$

6. Step 6 — Check: returning v_1p = 6.9000 ft/s to

   $$e = \dfrac{v_2' - v_1'}{v_1 - v_2}$$

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

Why the other options are there

• 13.8000 — kept a factor of two that cancels in the correct rearrangement.
• 3.4500 — dropped that same factor in the other direction.
• 7.5900 — rounded an intermediate value before the final step.

Reference: FE Handbook — Dynamics: Impact

Two billiard balls undergo a direct central impact on a table. Given velocity 1 before impact (v_1) = 29.5000 ft/s; velocity 2 before impact (v_2) = 17.5000 ft/s; velocity 1 after impact (v_1p) = 4.0000 ft/s; velocity 2 after impact (v_2p) = 16.0000 ft/s, determine the coefficient of restitution (e).

Start with the thinking

• The governing relation printed in this handbook section is Coefficient of restitution (direct central impact).
• Everything except e is given, so isolate e 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.
• Direct central impact between two bodies uses the coefficient of restitution to relate velocities before and after impact.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$e = \dfrac{v_2' - v_1'}{v_1 - v_2}$$

2. Step 2 — Rearrange symbolically for e:

   $$e = \dfrac{v_2' - v_1'}{v_1 - v_2}$$

3. Step 3 — List the givens: velocity 1 before impact (v_1) = 29.5000 ft/s, velocity 2 before impact (v_2) = 17.5000 ft/s, velocity 1 after impact (v_1p) = 4.0000 ft/s, velocity 2 after impact (v_2p) = 16.0000 ft/s.

4. Step 4 — Substitute the given values:

   $$e = \dfrac{v_2' - v_1'}{v_1 - v_2}$$

5. Step 5 — Evaluate:

   $$e = 1.0000$$

6. Step 6 — Check: returning e = 1.0000 to

   $$e = \dfrac{v_2' - v_1'}{v_1 - v_2}$$

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

Why the other options are there

• 2.0000 — kept a factor of two that cancels in the correct rearrangement.
• 0.5000 — dropped that same factor in the other direction.
• 1.1000 — rounded an intermediate value before the final step.

Reference: FE Handbook — Dynamics: Impact