Practice · Unit 5

Linear momentum and systems of particles exercises

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Try a few exercises and continue if you like

The page offers a short, varied set. There is no overall goal to complete.

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Linear momentum

Exercise to explore

Cyclist momentum

Type
Numerical
Difficulty
1/5
Time
5 min

A cyclist and bicycle are modeled as a 75 kg particle moving at +6.0 m/s. Calculate p_x.

Request a hint
  • Use the declared system, frame, and signs before substituting numbers.
Review the solution
  1. Principle

    Select the governing momentum relation and a consistent sign convention.

  2. Representation

    Represent every vector or flow component in the declared frame.

  3. Calculation

    Substitute the data, preserving signs and units.

Linear momentum

Exercise to explore

Momentum components

Type
Numerical
Difficulty
2/5
Time
5 min

A 2.0 kg particle has v=(3.0 i-4.0 j) m/s. Calculate p_x and p_y.

Request a hint
  • Use the declared system, frame, and signs before substituting numbers.
Review the solution
  1. Principle

    Select the governing momentum relation and a consistent sign convention.

  2. Representation

    Represent every vector or flow component in the declared frame.

  3. Calculation

    Substitute the data, preserving signs and units.

Linear momentum

Exercise to explore

Momentum magnitude

Type
Numerical
Difficulty
2/5
Time
5 min

A 5.0 kg object moves with velocity (2.0 i+1.5 j) m/s. Calculate |p|.

Request a hint
  • Use the declared system, frame, and signs before substituting numbers.
Review the solution
  1. Principle

    Select the governing momentum relation and a consistent sign convention.

  2. Representation

    Represent every vector or flow component in the declared frame.

  3. Calculation

    Substitute the data, preserving signs and units.

Linear momentum

Exercise to explore

Momentum change

Type
Numerical
Difficulty
2/5
Time
5 min

A 0.15 kg ball changes its 1D velocity from +20 m/s to -15 m/s. Calculate Δp_x.

Request a hint
  • Use the declared system, frame, and signs before substituting numbers.
Review the solution
  1. Principle

    Select the governing momentum relation and a consistent sign convention.

  2. Representation

    Represent every vector or flow component in the declared frame.

  3. Calculation

    Substitute the data, preserving signs and units.

Linear momentum

Exercise to explore

Same momentum

Type
Conceptual
Difficulty
2/5
Time
5 min

Two particles have the same nonzero momentum. A has greater mass. Which has greater speed?

Request a hint
  • Use the declared system, frame, and signs before substituting numbers.
Review the solution
  1. Principle

    Select the governing momentum relation and a consistent sign convention.

  2. Representation

    Represent every vector or flow component in the declared frame.

Impulse

Exercise to explore

Constant-force impulse

Type
Numerical
Difficulty
1/5
Time
5 min

A constant +x net force of 180 N acts for 0.12 s. Calculate J_x.

Request a hint
  • Use the declared system, frame, and signs before substituting numbers.
Review the solution
  1. Principle

    Select the governing momentum relation and a consistent sign convention.

  2. Representation

    Represent every vector or flow component in the declared frame.

  3. Calculation

    Substitute the data, preserving signs and units.

Impulse

Exercise to explore

Average impact force

Type
Numerical
Difficulty
2/5
Time
5 min

A 0.20 kg ball arrives downward at 8.0 m/s and rebounds upward at 6.0 m/s. Take +y upward. Contact lasts 0.040 s. Calculate average net force.

Request a hint
  • Use the declared system, frame, and signs before substituting numbers.
Review the solution
  1. Principle

    Select the governing momentum relation and a consistent sign convention.

  2. Representation

    Represent every vector or flow component in the declared frame.

  3. Calculation

    Substitute the data, preserving signs and units.

Impulse

Exercise to explore

Triangular impulse

Type
Numerical
Difficulty
2/5
Time
5 min

F_x(t) forms a positive triangle with base 0.30 s and height 120 N. Calculate impulse.

Request a hint
  • Use the declared system, frame, and signs before substituting numbers.
Review the solution
  1. Principle

    Select the governing momentum relation and a consistent sign convention.

  2. Representation

    Represent every vector or flow component in the declared frame.

  3. Calculation

    Substitute the data, preserving signs and units.

Impulse

Exercise to explore

Signed areas

Type
Numerical
Difficulty
3/5
Time
5 min

From 0 to 0.20 s, F_x=+50 N. From 0.20 to 0.50 s, F_x=-20 N. Calculate total J_x.

Request a hint
  • Use the declared system, frame, and signs before substituting numbers.
Review the solution
  1. Principle

    Select the governing momentum relation and a consistent sign convention.

  2. Representation

    Represent every vector or flow component in the declared frame.

  3. Calculation

    Substitute the data, preserving signs and units.

Impulse

Exercise to explore

Brief gravitational impulse

Type
Numerical
Difficulty
2/5
Time
5 min

Over 0.020 s a 0.50 kg ball also receives its weight's impulse. Take +y upward and g=9.8 m/s².

Request a hint
  • Use the declared system, frame, and signs before substituting numbers.
Review the solution
  1. Principle

    Select the governing momentum relation and a consistent sign convention.

  2. Representation

    Represent every vector or flow component in the declared frame.

  3. Calculation

    Substitute the data, preserving signs and units.

Conservation

Exercise to explore

Person–bag recoil

Type
Numerical
Difficulty
2/5
Time
5 min

A 70 kg person and 5.0 kg bag start at rest on ice. The bag leaves at +8.0 m/s. Calculate the person's velocity.

Request a hint
  • Use the declared system, frame, and signs before substituting numbers.
Review the solution
  1. Principle

    Select the governing momentum relation and a consistent sign convention.

  2. Representation

    Represent every vector or flow component in the declared frame.

  3. Calculation

    Substitute the data, preserving signs and units.

Conservation

Exercise to explore

Two fragments

Type
Numerical
Difficulty
2/5
Time
5 min

A resting 2.0 kg object separates into 0.50 kg and 1.50 kg fragments. The first leaves at +6.0 m/s. Calculate the second's velocity.

Request a hint
  • Use the declared system, frame, and signs before substituting numbers.
Review the solution
  1. Principle

    Select the governing momentum relation and a consistent sign convention.

  2. Representation

    Represent every vector or flow component in the declared frame.

  3. Calculation

    Substitute the data, preserving signs and units.

Conservation

Exercise to explore

External impulse

Type
Numerical
Difficulty
2/5
Time
5 min

A system initially has P_x=20 kg·m/s and receives J_x=-5.0 N·s. Calculate final P_x.

Request a hint
  • Use the declared system, frame, and signs before substituting numbers.
Review the solution
  1. Principle

    Select the governing momentum relation and a consistent sign convention.

  2. Representation

    Represent every vector or flow component in the declared frame.

  3. Calculation

    Substitute the data, preserving signs and units.

Conservation

Exercise to explore

Component conservation

Type
Conceptual
Difficulty
3/5
Time
5 min

During an interaction, horizontal external impulse is negligible but vertical external impulse exists. What may be approximately conserved?

Request a hint
  • Use the declared system, frame, and signs before substituting numbers.
Review the solution
  1. Principle

    Select the governing momentum relation and a consistent sign convention.

  2. Representation

    Represent every vector or flow component in the declared frame.

Conservation

Exercise to explore

Choosing the system

Type
Conceptual
Difficulty
2/5
Time
5 min

Two skaters push apart on ideal ice. The most useful system for direct horizontal momentum conservation is:

Request a hint
  • Use the declared system, frame, and signs before substituting numbers.
Review the solution
  1. Principle

    Select the governing momentum relation and a consistent sign convention.

  2. Representation

    Represent every vector or flow component in the declared frame.

Collisions

Exercise to explore

Sticking collision

Type
Numerical
Difficulty
2/5
Time
5 min

A 1200 kg car moves at +12 m/s and an 800 kg car at -5.0 m/s. They stick. Calculate v_f.

Request a hint
  • Use the declared system, frame, and signs before substituting numbers.
Review the solution
  1. Principle

    Select the governing momentum relation and a consistent sign convention.

  2. Representation

    Represent every vector or flow component in the declared frame.

  3. Calculation

    Substitute the data, preserving signs and units.

Collisions

Exercise to explore

Final energy after sticking

Type
Numerical
Difficulty
3/5
Time
5 min

Masses 2.0 kg and 1.0 kg move at +4.0 m/s and -2.0 m/s, then stick. Calculate final kinetic energy.

Request a hint
  • Use the declared system, frame, and signs before substituting numbers.
Review the solution
  1. Principle

    Select the governing momentum relation and a consistent sign convention.

  2. Representation

    Represent every vector or flow component in the declared frame.

  3. Calculation

    Substitute the data, preserving signs and units.

Collisions

Exercise to explore

Equal-mass elastic collision

Type
Conceptual
Difficulty
2/5
Time
5 min

m_1 arrives at +5 m/s and equal m_2 is at rest. What are the final velocities?

Request a hint
  • Use the declared system, frame, and signs before substituting numbers.
Review the solution
  1. Principle

    Select the governing momentum relation and a consistent sign convention.

  2. Representation

    Represent every vector or flow component in the declared frame.

Collisions

Exercise to explore

General elastic collision

Type
Numerical
Difficulty
3/5
Time
5 min

m_1=2.0 kg travels at +6.0 m/s and collides elastically with resting m_2=4.0 kg. Calculate v_1f and v_2f.

Request a hint
  • Use the declared system, frame, and signs before substituting numbers.
Review the solution
  1. Principle

    Select the governing momentum relation and a consistent sign convention.

  2. Representation

    Represent every vector or flow component in the declared frame.

  3. Calculation

    Substitute the data, preserving signs and units.

Collisions

Exercise to explore

Two-dimensional collision

Type
Numerical
Difficulty
3/5
Time
5 min

Two equal-mass balls: A arrives at +5.0 m/s on x and B rests. Afterward A moves at 3.0 m/s at +30°. Calculate v_Bx and v_By.

Request a hint
  • Use the declared system, frame, and signs before substituting numbers.
Review the solution
  1. Principle

    Select the governing momentum relation and a consistent sign convention.

  2. Representation

    Represent every vector or flow component in the declared frame.

  3. Calculation

    Substitute the data, preserving signs and units.

Collisions

Exercise to explore

Classifying a collision

Type
Conceptual
Difficulty
2/5
Time
5 min

In an isolated system, P is conserved but total K decreases. The collision is:

Request a hint
  • Use the declared system, frame, and signs before substituting numbers.
Review the solution
  1. Principle

    Select the governing momentum relation and a consistent sign convention.

  2. Representation

    Represent every vector or flow component in the declared frame.

Collisions

Exercise to explore

Insufficient 2D data

Type
Conceptual
Difficulty
3/5
Time
5 min

Four final components are unknown and only initial momentum is known. Is momentum conservation always sufficient?

Request a hint
  • Use the declared system, frame, and signs before substituting numbers.
Review the solution
  1. Principle

    Select the governing momentum relation and a consistent sign convention.

  2. Representation

    Represent every vector or flow component in the declared frame.

Center of mass

Exercise to explore

Center of mass in 1D

Type
Numerical
Difficulty
1/5
Time
5 min

A 2.0 kg mass is at x=0 and a 3.0 kg mass at x=5.0 m. Calculate x_cm.

Request a hint
  • Use the declared system, frame, and signs before substituting numbers.
Review the solution
  1. Principle

    Select the governing momentum relation and a consistent sign convention.

  2. Representation

    Represent every vector or flow component in the declared frame.

  3. Calculation

    Substitute the data, preserving signs and units.

Center of mass

Exercise to explore

Center of mass in 2D

Type
Numerical
Difficulty
2/5
Time
5 min

m_1=1 kg at (0,0), m_2=2 kg at (3,0), and m_3=1 kg at (0,4). Calculate x_cm,y_cm.

Request a hint
  • Use the declared system, frame, and signs before substituting numbers.
Review the solution
  1. Principle

    Select the governing momentum relation and a consistent sign convention.

  2. Representation

    Represent every vector or flow component in the declared frame.

  3. Calculation

    Substitute the data, preserving signs and units.

Center of mass

Exercise to explore

Center-of-mass velocity

Type
Numerical
Difficulty
2/5
Time
5 min

In 1D, m_1=2 kg has v_1=+4 m/s and m_2=3 kg has v_2=-1 m/s. Calculate v_cm.

Request a hint
  • Use the declared system, frame, and signs before substituting numbers.
Review the solution
  1. Principle

    Select the governing momentum relation and a consistent sign convention.

  2. Representation

    Represent every vector or flow component in the declared frame.

  3. Calculation

    Substitute the data, preserving signs and units.

Center of mass

Exercise to explore

Center-of-mass acceleration

Type
Numerical
Difficulty
2/5
Time
5 min

A constant-total-mass 5.0 kg system receives +10 N net external force. Calculate a_cm,x.

Request a hint
  • Use the declared system, frame, and signs before substituting numbers.
Review the solution
  1. Principle

    Select the governing momentum relation and a consistent sign convention.

  2. Representation

    Represent every vector or flow component in the declared frame.

  3. Calculation

    Substitute the data, preserving signs and units.

Center of mass

Exercise to explore

Center of a ring

Type
Conceptual
Difficulty
1/5
Time
5 min

The center of mass of an ideal ring:

Request a hint
  • Use the declared system, frame, and signs before substituting numbers.
Review the solution
  1. Principle

    Select the governing momentum relation and a consistent sign convention.

  2. Representation

    Represent every vector or flow component in the declared frame.

Center of mass

Exercise to explore

Center of mass after an explosion

Type
Conceptual
Difficulty
2/5
Time
5 min

A projectile explodes in flight. With no air resistance, what controls the center of mass?

Request a hint
  • Use the declared system, frame, and signs before substituting numbers.
Review the solution
  1. Principle

    Select the governing momentum relation and a consistent sign convention.

  2. Representation

    Represent every vector or flow component in the declared frame.

Variable mass

Exercise to explore

Sand leaking from a cart

Type
Conceptual
Difficulty
3/5
Time
5 min

Sand falls vertically from a resistance-free cart with the cart's instantaneous horizontal velocity. What ideally happens to cart speed?

Request a hint
  • Use the declared system, frame, and signs before substituting numbers.
Review the solution
  1. Principle

    Select the governing momentum relation and a consistent sign convention.

  2. Representation

    Represent every vector or flow component in the declared frame.

Variable mass

Exercise to explore

Sand on a conveyor

Type
Numerical
Difficulty
2/5
Time
5 min

Sand with zero initial horizontal velocity lands at 0.50 kg/s on a belt that carries it at +4.0 m/s. Calculate force on the sand.

Request a hint
  • Use the declared system, frame, and signs before substituting numbers.
Review the solution
  1. Principle

    Select the governing momentum relation and a consistent sign convention.

  2. Representation

    Represent every vector or flow component in the declared frame.

  3. Calculation

    Substitute the data, preserving signs and units.

Variable mass

Exercise to explore

Stopped water jet

Type
Numerical
Difficulty
2/5
Time
5 min

A 3.0 kg/s jet enters a plate at +8.0 m/s and exits with zero horizontal component. Calculate F_x on the water.

Request a hint
  • Use the declared system, frame, and signs before substituting numbers.
Review the solution
  1. Principle

    Select the governing momentum relation and a consistent sign convention.

  2. Representation

    Represent every vector or flow component in the declared frame.

  3. Calculation

    Substitute the data, preserving signs and units.

Variable mass

Exercise to explore

Flow turned 90°

Type
Numerical
Difficulty
3/5
Time
5 min

A 2.0 kg/s flow enters with v_in=(5,0) m/s and leaves with v_out=(0,5) m/s. Calculate F_x and F_y on the fluid.

Request a hint
  • Use the declared system, frame, and signs before substituting numbers.
Review the solution
  1. Principle

    Select the governing momentum relation and a consistent sign convention.

  2. Representation

    Represent every vector or flow component in the declared frame.

  3. Calculation

    Substitute the data, preserving signs and units.

Variable mass

Exercise to explore

Variable-mass boundary

Type
Conceptual
Difficulty
2/5
Time
5 min

What is the correct first step in a variable-mass problem?

Request a hint
  • Use the declared system, frame, and signs before substituting numbers.
Review the solution
  1. Principle

    Select the governing momentum relation and a consistent sign convention.

  2. Representation

    Represent every vector or flow component in the declared frame.

Variable mass

Exercise to explore

Momentum-flow units

Type
Conceptual
Difficulty
1/5
Time
5 min

The units of ṁΔv are:

Request a hint
  • Use the declared system, frame, and signs before substituting numbers.
Review the solution
  1. Principle

    Select the governing momentum relation and a consistent sign convention.

  2. Representation

    Represent every vector or flow component in the declared frame.

Propulsion

Exercise to explore

Engine thrust

Type
Numerical
Difficulty
1/5
Time
5 min

An engine expels mass at 4.0 kg/s with u_e=2500 m/s. Calculate ideal thrust magnitude.

Request a hint
  • Use the declared system, frame, and signs before substituting numbers.
Review the solution
  1. Principle

    Select the governing momentum relation and a consistent sign convention.

  2. Representation

    Represent every vector or flow component in the declared frame.

  3. Calculation

    Substitute the data, preserving signs and units.

Propulsion

Exercise to explore

Ideal Δv

Type
Numerical
Difficulty
2/5
Time
5 min

u_e=3000 m/s, m_i=1000 kg, and m_f=500 kg. Calculate ideal Δv.

Request a hint
  • Use the declared system, frame, and signs before substituting numbers.
Review the solution
  1. Principle

    Select the governing momentum relation and a consistent sign convention.

  2. Representation

    Represent every vector or flow component in the declared frame.

  3. Calculation

    Substitute the data, preserving signs and units.

Propulsion

Exercise to explore

Mass ratio

Type
Numerical
Difficulty
3/5
Time
5 min

A rocket needs Δv=1500 m/s with u_e=2500 m/s. Calculate m_i/m_f.

Request a hint
  • Use the declared system, frame, and signs before substituting numbers.
Review the solution
  1. Principle

    Select the governing momentum relation and a consistent sign convention.

  2. Representation

    Represent every vector or flow component in the declared frame.

  3. Calculation

    Substitute the data, preserving signs and units.

Propulsion

Exercise to explore

Approximate gravity loss

Type
Numerical
Difficulty
3/5
Time
5 min

With constant g and no drag, u_e=3000 m/s, m_i/m_f=2, and burn time 20 s. Calculate Δv=u_e ln(m_i/m_f) \Delta v=u_e\ln(m_i/m_f) -gΔt.

Request a hint
  • Use the declared system, frame, and signs before substituting numbers.
Review the solution
  1. Principle

    Select the governing momentum relation and a consistent sign convention.

  2. Representation

    Represent every vector or flow component in the declared frame.

  3. Calculation

    Substitute the data, preserving signs and units.

Propulsion

Exercise to explore

Rocket in vacuum

Type
Conceptual
Difficulty
1/5
Time
5 min

Why can a rocket accelerate in vacuum?

Request a hint
  • Use the declared system, frame, and signs before substituting numbers.
Review the solution
  1. Principle

    Select the governing momentum relation and a consistent sign convention.

  2. Representation

    Represent every vector or flow component in the declared frame.

Propulsion

Exercise to explore

Final mass fraction

Type
Numerical
Difficulty
3/5
Time
5 min

A rocket with u_e=3200 m/s needs Δv=2500 m/s. Calculate m_f/m_i.

Request a hint
  • Use the declared system, frame, and signs before substituting numbers.
Review the solution
  1. Principle

    Select the governing momentum relation and a consistent sign convention.

  2. Representation

    Represent every vector or flow component in the declared frame.

  3. Calculation

    Substitute the data, preserving signs and units.