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Newton-seconds are kilogram-meters per second

The impulse-momentum theorem sets an impulse equal to a momentum change. The numbers already matched: 8 N·s made 8 kg·m/s.

Can the newton-second and the kilogram-meter per second really be the same unit? The 8s say they must be.

Unpack the newton first.

A newton is the force that gives 1 kg an acceleration of 1 m/s², so 1 N = 1 kg·m/s².

Now build the newton-second.

1 N·s = 1 (kg·m/s²)·s.

The second you multiplied by cancels one of the two seconds under the line.

The unit chainsteplinethe newton in base units1 N = 1 kg·m/s²multiply by one second1 N·s = 1 (kg·m/s²)·scancel one second1 N·s = 1 kg·m/s
Three lines of unit algebra, and the two spellings meet.

1 N·s = 1 kg·m/s.

So the newton-second and the kilogram-meter per second are the same unit, exactly as the theorem requires.

One quantity, one unit, two spellings: N·s when you build it from force and time, kg·m/s when you build it from mass and velocity.

Where each spelling comes frombuilt fromspellingforce × timeN·smass × velocitykg·m/s
Build it from force and time, or from mass and velocity: the same unit either way.

Which unit is the same as the newton-second (N·s)?

Which chain of steps shows that the newton-second is the kilogram-meter per second?

The newton-second and the kilogram-meter per second are the same unit: 1 N·s = 1 kg·m/s.

Show that the newton-second is the same unit as the kilogram-meter per second.

Write down the newton in base units:

1 N = 1 kg·m/s²

Multiply by one second:

1 N·s = 1 (kg·m/s²)·s

Cancel one second:

1 N·s = 1 (kg·m/s²)·s = 1 kg·m/s

The newton-second and the kilogram-meter per second are the same unit.

A hammer blow delivers an impulse of 9 N·s to a tent stake. State the stake's momentum change, in either spelling of the unit.

The theorem: the impulse equals the momentum change.

The units agree: 1 N·s = 1 kg·m/s.

So the number carries straight over.

The stake's momentum change is 9 kg·m/s, which is the same as 9 N·s.

A dart receives an impulse of 2.2 N·s. Enter its momentum change in kg·m/s.

Your turn

Which unit can state a momentum change?

Gunnar writes four unit chains. Which one is correct?

Impulse and momentum change share one unit. The newton-second is one spelling of it. Type the other spelling, built from the mass and velocity units.