Linear Momentum

Dynamics · FE Reference Handbook section

Dynamics

3 formulas

10 exam-style examples

~51 min

All Dynamics lectures

Handbook notes for this section

Definitions and conditions exactly as the handbook states them.

Assuming constant mass, the equation of motion of a particle may be written as
For a system of particles, by integrating and summing over the number of particles, this may be expanded to
The term on the left side of the equation is the linear momentum of a system of particles at time t2. The first term on the right
side of the equation is the linear momentum of a system of particles at time t1. The second term on the right side of the equation
is the impulse of the force F from time t1 to t2. It should be noted that the above equation is a vector equation. Component scalar
equations may be obtained by considering the momentum and force in a set of orthogonal directions.

Core formulas for this FE topic

Definitions, applicability, units, assumptions and worked examples for each relation.

Worked exam-style examples

The four ways this section is written on the real exam — thoughts first, then equations, then substitution.

Example 1

Linear momentum — solve for linear momentum — Linear Momentum

A rail car's linear momentum is computed before a coupling collision. Given mass (m) = 4.1000 slug; velocity (v) = 7.5000 ft/s, determine the linear momentum (p) in slug-ft/s.

Given

$mass (m) = 4.1000 slug$
$velocity (v) = 7.5000 ft/s$

Find

linear momentum (p), in slug-ft/s

Start with the thinking

The governing relation printed in this handbook section is Linear momentum.
Everything except p is given, so isolate p 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.
Linear momentum of a particle equals mass times velocity and is conserved in the absence of external impulse.

Figure 1 — schematic for Linear momentum — solve for linear momentum — Linear Momentum

Step-by-step solution

Step 1 — State the governing relation:
$p = m v$
Step 2 — Rearrange symbolically for p:
$p = m v$
Step 3
$List the givens: mass (m) = 4.1000 slug, velocity (v) = 7.5000 ft/s$
Step 4 — Substitute the given values:
$p = 4.1000 7.5000$
Step 5 — Evaluate:
$p = 30.7500\ \text{slug-ft/s}$
Step 6 — Check: returning p = 30.7500 slug-ft/s to
$p = m v$
reproduces the given quantities, and both sides carry the same units.

Answer:

p = 30.7500\ \text{slug-ft/s}

Why the other options are there

61.5000 — kept a factor of two that cancels in the correct rearrangement.
15.3750 — dropped that same factor in the other direction.
33.8250 — rounded an intermediate value before the final step.

Reference: FE Handbook — Dynamics: Linear Momentum

Example 2

Linear momentum — solve for mass — Linear Momentum (2)

A thrown ball carries linear momentum used in an impulse-momentum problem. Given velocity (v) = 8.0000 ft/s; linear momentum (p) = 1,223 slug-ft/s, determine the mass (m) in slug.

Given

$velocity (v) = 8.0000 ft/s$
$linear momentum (p) = 1,223 slug-ft/s$

Find

mass (m), in slug

Start with the thinking

The governing relation printed in this handbook section is Linear momentum.
Everything except m is given, so isolate m 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.
Linear momentum of a particle equals mass times velocity and is conserved in the absence of external impulse.

Figure 2 — schematic for Linear momentum — solve for mass — Linear Momentum (2)

Step-by-step solution

Step 1 — State the governing relation:
$p = m v$
Step 2 — Rearrange symbolically for m:
$m = \dfrac{p}{v}$
Step 3
$List the givens: velocity (v) = 8.0000 ft/s, linear momentum (p) = 1,223 slug-ft/s$
Step 4 — Substitute the given values:
$m = \dfrac{1223}{8.0000}$
Step 5 — Evaluate:
$m = 152.9\ \text{slug}$
Step 6 — Check: returning m = 152.9 slug to
$p = m v$
reproduces the given quantities, and both sides carry the same units.

Answer:

m = 152.9\ \text{slug}

Why the other options are there

305.8 — kept a factor of two that cancels in the correct rearrangement.
76.4375 — dropped that same factor in the other direction.
168.2 — rounded an intermediate value before the final step.

Reference: FE Handbook — Dynamics: Linear Momentum

Example 3

Linear momentum — solve for velocity — Linear Momentum (3)

A rocket sled's linear momentum changes as thrust is applied. Given mass (m) = 7.6000 slug; linear momentum (p) = 2,083 slug-ft/s, determine the velocity (v) in ft/s.

Given

$mass (m) = 7.6000 slug$
$linear momentum (p) = 2,083 slug-ft/s$

Find

velocity (v), in ft/s

Start with the thinking

The governing relation printed in this handbook section is Linear momentum.
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.
Linear momentum of a particle equals mass times velocity and is conserved in the absence of external impulse.

Figure 3 — schematic for Linear momentum — solve for velocity — Linear Momentum (3)

Step-by-step solution

Step 1 — State the governing relation:
$p = m v$
Step 2 — Rearrange symbolically for v:
$v = \dfrac{p}{m}$
Step 3
$List the givens: mass (m) = 7.6000 slug, linear momentum (p) = 2,083 slug-ft/s$
Step 4 — Substitute the given values:
$v = \dfrac{2083}{7.6000}$
Step 5 — Evaluate:
$v = 274.1\ \text{ft/s}$
Step 6 — Check: returning v = 274.1 ft/s to
$p = m v$
reproduces the given quantities, and both sides carry the same units.

Answer:

v = 274.1\ \text{ft/s}

Why the other options are there

548.2 — kept a factor of two that cancels in the correct rearrangement.
137.0 — dropped that same factor in the other direction.
301.5 — rounded an intermediate value before the final step.

Reference: FE Handbook — Dynamics: Linear Momentum

Example 4

Linear momentum — solve for linear momentum (case 2) — Linear Momentum (4)

A rail car's linear momentum is computed before a coupling collision. Given mass (m) = 30.7000 slug; velocity (v) = 13.0000 ft/s, determine the linear momentum (p) in slug-ft/s.

Given

$mass (m) = 30.7000 slug$
$velocity (v) = 13.0000 ft/s$

Find

linear momentum (p), in slug-ft/s

Start with the thinking

The governing relation printed in this handbook section is Linear momentum.
Everything except p is given, so isolate p 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.
Linear momentum of a particle equals mass times velocity and is conserved in the absence of external impulse.

Figure 4 — schematic for Linear momentum — solve for linear momentum (case 2) — Linear Momentum (4)

Step-by-step solution

Step 1 — State the governing relation:
$p = m v$
Step 2 — Rearrange symbolically for p:
$p = m v$
Step 3
$List the givens: mass (m) = 30.7000 slug, velocity (v) = 13.0000 ft/s$
Step 4 — Substitute the given values:
$p = 30.7000 13.0000$
Step 5 — Evaluate:
$p = 399.1\ \text{slug-ft/s}$
Step 6 — Check: returning p = 399.1 slug-ft/s to
$p = m v$
reproduces the given quantities, and both sides carry the same units.

Answer:

p = 399.1\ \text{slug-ft/s}

Why the other options are there

798.2 — kept a factor of two that cancels in the correct rearrangement.
199.5 — dropped that same factor in the other direction.
439.0 — rounded an intermediate value before the final step.

Reference: FE Handbook — Dynamics: Linear Momentum

Example 5

Linear momentum — solve for mass (case 2) — Linear Momentum (5)

A thrown ball carries linear momentum used in an impulse-momentum problem. Given velocity (v) = 5.5000 ft/s; linear momentum (p) = 484.0 slug-ft/s, determine the mass (m) in slug.

Given

$velocity (v) = 5.5000 ft/s$
$linear momentum (p) = 484.0 slug-ft/s$

Find

mass (m), in slug

Start with the thinking

The governing relation printed in this handbook section is Linear momentum.
Everything except m is given, so isolate m 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.
Linear momentum of a particle equals mass times velocity and is conserved in the absence of external impulse.

Figure 5 — schematic for Linear momentum — solve for mass (case 2) — Linear Momentum (5)

Step-by-step solution

Step 1 — State the governing relation:
$p = m v$
Step 2 — Rearrange symbolically for m:
$m = \dfrac{p}{v}$
Step 3
$List the givens: velocity (v) = 5.5000 ft/s, linear momentum (p) = 484.0 slug-ft/s$
Step 4 — Substitute the given values:
$m = \dfrac{484.0}{5.5000}$
Step 5 — Evaluate:
$m = 88.0000\ \text{slug}$
Step 6 — Check: returning m = 88.0000 slug to
$p = m v$
reproduces the given quantities, and both sides carry the same units.

Answer:

m = 88.0000\ \text{slug}

Why the other options are there

176.0 — kept a factor of two that cancels in the correct rearrangement.
44.0000 — dropped that same factor in the other direction.
96.8000 — rounded an intermediate value before the final step.

Reference: FE Handbook — Dynamics: Linear Momentum

Example 6

Linear momentum — solve for velocity (case 2) — Linear Momentum (6)

A rocket sled's linear momentum changes as thrust is applied. Given mass (m) = 24.3000 slug; linear momentum (p) = 2,130 slug-ft/s, determine the velocity (v) in ft/s.

Given

$mass (m) = 24.3000 slug$
$linear momentum (p) = 2,130 slug-ft/s$

Find

velocity (v), in ft/s

Start with the thinking

The governing relation printed in this handbook section is Linear momentum.
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.
Linear momentum of a particle equals mass times velocity and is conserved in the absence of external impulse.

Figure 6 — schematic for Linear momentum — solve for velocity (case 2) — Linear Momentum (6)

Step-by-step solution

Step 1 — State the governing relation:
$p = m v$
Step 2 — Rearrange symbolically for v:
$v = \dfrac{p}{m}$
Step 3
$List the givens: mass (m) = 24.3000 slug, linear momentum (p) = 2,130 slug-ft/s$
Step 4 — Substitute the given values:
$v = \dfrac{2130}{24.3000}$
Step 5 — Evaluate:
$v = 87.6543\ \text{ft/s}$
Step 6 — Check: returning v = 87.6543 ft/s to
$p = m v$
reproduces the given quantities, and both sides carry the same units.

Answer:

v = 87.6543\ \text{ft/s}

Why the other options are there

175.3 — kept a factor of two that cancels in the correct rearrangement.
43.8272 — dropped that same factor in the other direction.
96.4198 — rounded an intermediate value before the final step.

Reference: FE Handbook — Dynamics: Linear Momentum

Example 7

Linear momentum — solve for linear momentum (case 3) — Linear Momentum (7)

A rail car's linear momentum is computed before a coupling collision. Given mass (m) = 24.7000 slug; velocity (v) = 82.5000 ft/s, determine the linear momentum (p) in slug-ft/s.

Given

$mass (m) = 24.7000 slug$
$velocity (v) = 82.5000 ft/s$

Find

linear momentum (p), in slug-ft/s

Start with the thinking

The governing relation printed in this handbook section is Linear momentum.
Everything except p is given, so isolate p 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.
Linear momentum of a particle equals mass times velocity and is conserved in the absence of external impulse.

Figure 7 — schematic for Linear momentum — solve for linear momentum (case 3) — Linear Momentum (7)

Step-by-step solution

Step 1 — State the governing relation:
$p = m v$
Step 2 — Rearrange symbolically for p:
$p = m v$
Step 3
$List the givens: mass (m) = 24.7000 slug, velocity (v) = 82.5000 ft/s$
Step 4 — Substitute the given values:
$p = 24.7000 82.5000$
Step 5 — Evaluate:
$p = 2038\ \text{slug-ft/s}$
Step 6 — Check: returning p = 2,038 slug-ft/s to
$p = m v$
reproduces the given quantities, and both sides carry the same units.

Answer:

p = 2038\ \text{slug-ft/s}

Why the other options are there

4,076 — kept a factor of two that cancels in the correct rearrangement.
1,019 — dropped that same factor in the other direction.
2,242 — rounded an intermediate value before the final step.

Reference: FE Handbook — Dynamics: Linear Momentum

Example 8

Linear momentum — solve for mass (case 3) — Linear Momentum (8)

A thrown ball carries linear momentum used in an impulse-momentum problem. Given velocity (v) = 55.5000 ft/s; linear momentum (p) = 2,675 slug-ft/s, determine the mass (m) in slug.

Given

$velocity (v) = 55.5000 ft/s$
$linear momentum (p) = 2,675 slug-ft/s$

Find

mass (m), in slug

Start with the thinking

The governing relation printed in this handbook section is Linear momentum.
Everything except m is given, so isolate m 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.
Linear momentum of a particle equals mass times velocity and is conserved in the absence of external impulse.

Figure 8 — schematic for Linear momentum — solve for mass (case 3) — Linear Momentum (8)

Step-by-step solution

Step 1 — State the governing relation:
$p = m v$
Step 2 — Rearrange symbolically for m:
$m = \dfrac{p}{v}$
Step 3
$List the givens: velocity (v) = 55.5000 ft/s, linear momentum (p) = 2,675 slug-ft/s$
Step 4 — Substitute the given values:
$m = \dfrac{2675}{55.5000}$
Step 5 — Evaluate:
$m = 48.1982\ \text{slug}$
Step 6 — Check: returning m = 48.1982 slug to
$p = m v$
reproduces the given quantities, and both sides carry the same units.

Answer:

m = 48.1982\ \text{slug}

Why the other options are there

96.3964 — kept a factor of two that cancels in the correct rearrangement.
24.0991 — dropped that same factor in the other direction.
53.0180 — rounded an intermediate value before the final step.

Reference: FE Handbook — Dynamics: Linear Momentum

Example 9

Linear momentum — solve for velocity (case 3) — Linear Momentum (9)

A rocket sled's linear momentum changes as thrust is applied. Given mass (m) = 20.4000 slug; linear momentum (p) = 2,746 slug-ft/s, determine the velocity (v) in ft/s.

Given

$mass (m) = 20.4000 slug$
$linear momentum (p) = 2,746 slug-ft/s$

Find

velocity (v), in ft/s

Start with the thinking

The governing relation printed in this handbook section is Linear momentum.
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.
Linear momentum of a particle equals mass times velocity and is conserved in the absence of external impulse.

Figure 9 — schematic for Linear momentum — solve for velocity (case 3) — Linear Momentum (9)

Step-by-step solution

Step 1 — State the governing relation:
$p = m v$
Step 2 — Rearrange symbolically for v:
$v = \dfrac{p}{m}$
Step 3
$List the givens: mass (m) = 20.4000 slug, linear momentum (p) = 2,746 slug-ft/s$
Step 4 — Substitute the given values:
$v = \dfrac{2746}{20.4000}$
Step 5 — Evaluate:
$v = 134.6\ \text{ft/s}$
Step 6 — Check: returning v = 134.6 ft/s to
$p = m v$
reproduces the given quantities, and both sides carry the same units.

Answer:

v = 134.6\ \text{ft/s}

Why the other options are there

269.2 — kept a factor of two that cancels in the correct rearrangement.
67.3039 — dropped that same factor in the other direction.
148.1 — rounded an intermediate value before the final step.

Reference: FE Handbook — Dynamics: Linear Momentum

Example 10

Linear momentum — solve for linear momentum (case 4) — Linear Momentum (10)

A rail car's linear momentum is computed before a coupling collision. Given mass (m) = 12.1000 slug; velocity (v) = 38.5000 ft/s, determine the linear momentum (p) in slug-ft/s.

Given

$mass (m) = 12.1000 slug$
$velocity (v) = 38.5000 ft/s$

Find

linear momentum (p), in slug-ft/s

Start with the thinking

The governing relation printed in this handbook section is Linear momentum.
Everything except p is given, so isolate p 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.
Linear momentum of a particle equals mass times velocity and is conserved in the absence of external impulse.

Figure 10 — schematic for Linear momentum — solve for linear momentum (case 4) — Linear Momentum (10)

Step-by-step solution

Step 1 — State the governing relation:
$p = m v$
Step 2 — Rearrange symbolically for p:
$p = m v$
Step 3
$List the givens: mass (m) = 12.1000 slug, velocity (v) = 38.5000 ft/s$
Step 4 — Substitute the given values:
$p = 12.1000 38.5000$
Step 5 — Evaluate:
$p = 465.8\ \text{slug-ft/s}$
Step 6 — Check: returning p = 465.8 slug-ft/s to
$p = m v$
reproduces the given quantities, and both sides carry the same units.

Answer:

p = 465.8\ \text{slug-ft/s}

Why the other options are there

931.7 — kept a factor of two that cancels in the correct rearrangement.
232.9 — dropped that same factor in the other direction.
512.4 — rounded an intermediate value before the final step.

Reference: FE Handbook — Dynamics: Linear Momentum

Linear Momentum

Handbook notes for this section

Core formulas for this FE topic

mdv/dt=Fmdv/dt = Fmdv/dt=F

mdv=Fdtmdv = Fdtmdv=Fdt

Rmi ^viht 2 = Rmi ^viht1 + R # Fi dtNeeds technical review

Worked exam-style examples