A dynamics problem uses Constant-acceleration kinematics. Given initial velocity (v0) = 83.0000 ft/s; acceleration (a) = -1.5000 ft/s²; distance (s) = 426.0 ft, determine the final velocity (v) in ft/s.

Start with the thinking

• The governing relation printed in this handbook section is Constant-acceleration kinematics.
• Everything except v is given, so isolate v 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.
• Dynamics items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$v^2 = v_0^2 + 2 a s$$

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

3. Step 3 — List the givens: initial velocity (v0) = 83.0000 ft/s, acceleration (a) = -1.5000 ft/s², distance (s) = 426.0 ft.

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

5. Step 5 — Evaluate:

   $$v = 74.9066\ \text{ft/s}$$

6. Step 6 — Check: returning v = 74.9066 ft/s to

   $$v^2 = v_0^2 + 2 a s$$

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

Why the other options are there

• 149.8 — kept a factor of two that cancels in the correct rearrangement.
• 37.4533 — dropped that same factor in the other direction.
• 82.3973 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Dynamics → Constant Acceleration

An elevator speeds up with constant acceleration before reaching cruise speed. Given initial velocity (v_0) = 43.5000 ft/s; constant acceleration (a) = 13.9000 ft/s^2; time (t) = 9.9000 s, determine the final velocity (v) in ft/s.

Start with the thinking

• The governing relation printed in this handbook section is Constant acceleration.
• Everything except v is given, so isolate v 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.
• Constant acceleration kinematics relates initial velocity, final velocity, acceleration, and time for a particle.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$v = v_0 + a t$$

2. Step 2 — Rearrange symbolically for v:

   $$v = v_0 + a t$$

3. Step 3 — List the givens: initial velocity (v_0) = 43.5000 ft/s, constant acceleration (a) = 13.9000 ft/s^2, time (t) = 9.9000 s.

4. Step 4 — Substitute the given values:

   $$v = v_0 + 13.9000 9.9000$$

5. Step 5 — Evaluate:

   $$v = 181.1\ \text{ft/s}$$

6. Step 6 — Check: returning v = 181.1 ft/s to

   $$v = v_0 + a t$$

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

Why the other options are there

• 362.2 — kept a factor of two that cancels in the correct rearrangement.
• 90.5550 — dropped that same factor in the other direction.
• 199.2 — rounded an intermediate value before the final step.

Reference: FE Handbook — Dynamics: Constant Acceleration

A dynamics problem uses Constant-acceleration kinematics. Given initial velocity (v0) = 45.0000 ft/s; distance (s) = 58.0000 ft; final velocity (v) = 112.5 ft/s, determine the acceleration (a) in ft/s².

Start with the thinking

• The governing relation printed in this handbook section is Constant-acceleration kinematics.
• Everything except a is given, so isolate a 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.
• Dynamics items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$v^2 = v_0^2 + 2 a s$$

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

3. Step 3 — List the givens: initial velocity (v0) = 45.0000 ft/s, distance (s) = 58.0000 ft, final velocity (v) = 112.5 ft/s.

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

5. Step 5 — Evaluate:

   $$a = 91.6487\ \text{ft/s²}$$

6. Step 6 — Check: returning a = 91.6487 ft/s² to

   $$v^2 = v_0^2 + 2 a s$$

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

Why the other options are there

• 183.3 — kept a factor of two that cancels in the correct rearrangement.
• 45.8244 — dropped that same factor in the other direction.
• 100.8 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Dynamics → Constant Acceleration

A dropped object near Earth's surface undergoes constant acceleration due to gravity. Given constant acceleration (a) = 31.7000 ft/s^2; time (t) = 4.2000 s; final velocity (v) = 143.3 ft/s, determine the initial velocity (v_0) in ft/s.

Start with the thinking

• The governing relation printed in this handbook section is Constant acceleration.
• Everything except v_0 is given, so isolate v_0 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.
• Constant acceleration kinematics relates initial velocity, final velocity, acceleration, and time for a particle.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$v = v_0 + a t$$

2. Step 2 — Rearrange symbolically for v_0:

   $$v_{0} = v - a t$$

3. Step 3 — List the givens: constant acceleration (a) = 31.7000 ft/s^2, time (t) = 4.2000 s, final velocity (v) = 143.3 ft/s.

4. Step 4 — Substitute the given values:

   $$v_{0} = 143.3 - 31.7000 4.2000$$

5. Step 5 — Evaluate:

   $$v_{0} = 10.1600\ \text{ft/s}$$

6. Step 6 — Check: returning v_0 = 10.1600 ft/s to

   $$v = v_0 + a t$$

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

Why the other options are there

• 20.3200 — kept a factor of two that cancels in the correct rearrangement.
• 5.0800 — dropped that same factor in the other direction.
• 11.1760 — rounded an intermediate value before the final step.

Reference: FE Handbook — Dynamics: Constant Acceleration

A dynamics problem uses Constant-acceleration kinematics. Given initial velocity (v0) = 93.0000 ft/s; acceleration (a) = -14.5000 ft/s²; final velocity (v) = 81.0000 ft/s, determine the distance (s) in ft.

Start with the thinking

• The governing relation printed in this handbook section is Constant-acceleration kinematics.
• Everything except s is given, so isolate s 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.
• Dynamics items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$v^2 = v_0^2 + 2 a s$$

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

3. Step 3 — List the givens: initial velocity (v0) = 93.0000 ft/s, acceleration (a) = -14.5000 ft/s², final velocity (v) = 81.0000 ft/s.

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

5. Step 5 — Evaluate:

   $$s = 72.0000\ \text{ft}$$

6. Step 6 — Check: returning s = 72.0000 ft to

   $$v^2 = v_0^2 + 2 a s$$

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

Why the other options are there

• 144.0 — kept a factor of two that cancels in the correct rearrangement.
• 36.0000 — dropped that same factor in the other direction.
• 79.2000 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Dynamics → Constant Acceleration

A car accelerates from a stoplight under constant acceleration. Given initial velocity (v_0) = 32.5000 ft/s; time (t) = 6.0000 s; final velocity (v) = 193.2 ft/s, determine the constant acceleration (a) in ft/s^2.

Start with the thinking

• The governing relation printed in this handbook section is Constant acceleration.
• Everything except a is given, so isolate a 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.
• Constant acceleration kinematics relates initial velocity, final velocity, acceleration, and time for a particle.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$v = v_0 + a t$$

2. Step 2 — Rearrange symbolically for a:

   $$a = \dfrac{v - v_0}{t}$$

3. Step 3 — List the givens: initial velocity (v_0) = 32.5000 ft/s, time (t) = 6.0000 s, final velocity (v) = 193.2 ft/s.

4. Step 4 — Substitute the given values:

   $$a = \dfrac{193.2 - v_0}{6.0000}$$

5. Step 5 — Evaluate:

   a = 26.7833\ \text{ft/s^2}

6. Step 6 — Check: returning a = 26.7833 ft/s^2 to

   $$v = v_0 + a t$$

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

Why the other options are there

• 53.5667 — kept a factor of two that cancels in the correct rearrangement.
• 13.3917 — dropped that same factor in the other direction.
• 29.4617 — rounded an intermediate value before the final step.

Reference: FE Handbook — Dynamics: Constant Acceleration

A dynamics problem uses Constant-acceleration kinematics. Given initial velocity (v0) = 23.0000 ft/s; acceleration (a) = 27.0000 ft/s²; distance (s) = 372.0 ft, determine the final velocity (v) in ft/s.

Start with the thinking

• The governing relation printed in this handbook section is Constant-acceleration kinematics.
• Everything except v is given, so isolate v 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.
• Dynamics items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$v^2 = v_0^2 + 2 a s$$

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

3. Step 3 — List the givens: initial velocity (v0) = 23.0000 ft/s, acceleration (a) = 27.0000 ft/s², distance (s) = 372.0 ft.

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

5. Step 5 — Evaluate:

   $$v = 143.6\ \text{ft/s}$$

6. Step 6 — Check: returning v = 143.6 ft/s to

   $$v^2 = v_0^2 + 2 a s$$

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

Why the other options are there

• 287.2 — kept a factor of two that cancels in the correct rearrangement.
• 71.7931 — dropped that same factor in the other direction.
• 157.9 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Dynamics → Constant Acceleration

An elevator speeds up with constant acceleration before reaching cruise speed. Given initial velocity (v_0) = 51.5000 ft/s; constant acceleration (a) = 19.2000 ft/s^2; final velocity (v) = 369.9 ft/s, determine the time (t) in s.

Start with the thinking

• The governing relation printed in this handbook section is Constant acceleration.
• Everything except t is given, so isolate t 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.
• Constant acceleration kinematics relates initial velocity, final velocity, acceleration, and time for a particle.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$v = v_0 + a t$$

2. Step 2 — Rearrange symbolically for t:

   $$t = \dfrac{v - v_0}{a}$$

3. Step 3 — List the givens: initial velocity (v_0) = 51.5000 ft/s, constant acceleration (a) = 19.2000 ft/s^2, final velocity (v) = 369.9 ft/s.

4. Step 4 — Substitute the given values:

   $$t = \dfrac{369.9 - v_0}{19.2000}$$

5. Step 5 — Evaluate:

   $$t = 16.5833\ \text{s}$$

6. Step 6 — Check: returning t = 16.5833 s to

   $$v = v_0 + a t$$

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

Why the other options are there

• 33.1667 — kept a factor of two that cancels in the correct rearrangement.
• 8.2917 — dropped that same factor in the other direction.
• 18.2417 — rounded an intermediate value before the final step.

Reference: FE Handbook — Dynamics: Constant Acceleration

A dynamics problem uses Constant-acceleration kinematics. Given initial velocity (v0) = 78.0000 ft/s; distance (s) = 367.0 ft; final velocity (v) = 9.0000 ft/s, determine the acceleration (a) in ft/s².

Start with the thinking

• The governing relation printed in this handbook section is Constant-acceleration kinematics.
• Everything except a is given, so isolate a 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.
• Dynamics items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$v^2 = v_0^2 + 2 a s$$

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

3. Step 3 — List the givens: initial velocity (v0) = 78.0000 ft/s, distance (s) = 367.0 ft, final velocity (v) = 9.0000 ft/s.

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

5. Step 5 — Evaluate:

   $$a = -8.1785\ \text{ft/s²}$$

6. Step 6 — Check: returning a = -8.1785 ft/s² to

   $$v^2 = v_0^2 + 2 a s$$

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

Why the other options are there

• -16.3569 — kept a factor of two that cancels in the correct rearrangement.
• -4.0892 — dropped that same factor in the other direction.
• -8.9963 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Dynamics → Constant Acceleration

A dropped object near Earth's surface undergoes constant acceleration due to gravity. Given initial velocity (v_0) = 35.5000 ft/s; constant acceleration (a) = 7.0000 ft/s^2; time (t) = 1.9000 s, determine the final velocity (v) in ft/s.

Start with the thinking

• The governing relation printed in this handbook section is Constant acceleration.
• Everything except v is given, so isolate v 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.
• Constant acceleration kinematics relates initial velocity, final velocity, acceleration, and time for a particle.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$v = v_0 + a t$$

2. Step 2 — Rearrange symbolically for v:

   $$v = v_0 + a t$$

3. Step 3 — List the givens: initial velocity (v_0) = 35.5000 ft/s, constant acceleration (a) = 7.0000 ft/s^2, time (t) = 1.9000 s.

4. Step 4 — Substitute the given values:

   $$v = v_0 + 7.0000 1.9000$$

5. Step 5 — Evaluate:

   $$v = 48.8000\ \text{ft/s}$$

6. Step 6 — Check: returning v = 48.8000 ft/s to

   $$v = v_0 + a t$$

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

Why the other options are there

• 97.6000 — kept a factor of two that cancels in the correct rearrangement.
• 24.4000 — dropped that same factor in the other direction.
• 53.6800 — rounded an intermediate value before the final step.

Reference: FE Handbook — Dynamics: Constant Acceleration