Unit 5 · Topic 02

Impulse and momentum change

Impulse measures the accumulated effect of net force over time. On a force-time graph it is algebraic area, not absolute geometric area.

Unit 5ImpulseOpen navigation

Concept 01

Area changes momentum

Essential The minimum you should retain

Net-force impulse equals momentum change: J=Δp \vec J=\Delta\vec p .

UnderstandInterpret and connect

A small force over a long time can give the same impulse as a large force over a short time.

DeepenFormulation and conditions

For a variable force, J=∫F_net dt J=\int F_{net}dt , with signed area for each component.

ExploreConnections for further study

The theorem connects force measurements during impacts to states before and after.

Worked example

Impulse from a triangular force

F_x rises from 0 to 120 N in 0.15 s and returns to 0 over another 0.15 s.

Given
  • base=0.30 s
  • height=120 N
Target

Calculate total impulse.

  1. Principle

    J is area under F(t).

  2. Geometry

    The full pulse is a triangle.

  3. Calculation

    J=(1/2)(0.30)(120)=18 N·s.

  4. Interpretation

    Δp_x=18 kg·m/s.

Conclusion

Impulse is 18 N·s.

Mathematical relation

Impulse and momentum change

J→=∫titfF→dt=Δp→ \vec J=\int_{t_i}^{t_f}\vec F_{net}dt=\Delta\vec p
Represents

Accumulated effect of net force.

Physical interpretation

Signed area under F(t) gives Δp.

DeepenVariables, conditions, and checks

Variables

J
impulse; usual unit: N·s
F_net
net force; usual unit: N

Conditions of application

  • F and p use the same inertial frame.

Dimensional check

N·s=kg·m/s.

Impulse is signed area

Force rises linearly and returns to zero.
  • F_x(t)

The positive triangle has base 0.30 s, height 120 N, and area 18 N·s.

Concept 02

A useful average force

Essential The minimum you should retain

Average force reproduces the same impulse over the same interval.

UnderstandInterpret and connect

F_avg=Δp/Δt and need not equal the peak of a variable force.

DeepenFormulation and conditions

The vector equality preserves signs and directions component by component.

ExploreConnections for further study

Increasing contact time can reduce average force for a fixed momentum change.

Mathematical relation

Average net force

F→=Δp→Δt \vec F_{med}=\Delta\vec p/\Delta t
Represents

Constant force equivalent in impulse.

Physical interpretation

For fixed Δp, longer time means lower average force.

DeepenVariables, conditions, and checks

Variables

F_med
average net force; usual unit: N
Δt
duration; usual unit: s

Conditions of application

  • Δt>0.

Dimensional check

(kg·m/s)/s=N.

Equal area, different average force

Two equal-area rectangles compare force and duration.
  • 100 N for 0.2 s
  • 50 N for 0.4 s

Both pulses give the same impulse; the longer one needs a smaller height.

Concept 03

Above and below the axis

Essential The minimum you should retain

A region of F_x(t) above the axis contributes positive impulse and one below contributes negative impulse.

UnderstandInterpret and connect

Opposite contributions can cancel partly or completely.

DeepenFormulation and conditions

Integrating the absolute value answers a different question and overestimates net Δp_x.

ExploreConnections for further study

Discrete sensor data can approximate impulse with signed trapezoids.

Concept 04

Modeling a short interval

Essential The minimum you should retain

During a collision, intense contact forces act alongside external forces such as weight.

UnderstandInterpret and connect

If an external impulse is small compared with contact impulse, it may be neglected as a stated approximation.

DeepenFormulation and conditions

The decision is based on comparing impulses, not only instantaneous forces.

ExploreConnections for further study

This criterion explains why some momentum components are approximately conserved in brief collisions.

Concept review

Common errors

Each warning includes a concrete way to review the reasoning, not only an incorrect-answer marker.

Using J=FΔt with any value of a variable force.

Use J=∫Fdt or a correctly defined average force.

Adding absolute areas under F(t).

Component impulse is algebraic area.