Impulse Calculator

Impulse is force held over time

Impulse is how much push an object receives, counted as the force multiplied by the time the force lasts. Lean on a shopping cart for 2 seconds with 50 newtons and you have delivered 100 newton-seconds. Tap the same cart for 0.1 second with the same 50 newtons and you have delivered 5 newton-seconds. The cart answers to the size of the delivery, not to how hard your hand felt during it.

The NASA Glenn Research Center page on momentum frames the same relation through Newton's second law: a force changes momentum at a rate equal to the force, so force summed over time gives the change in momentum. That is the impulse-momentum theorem, written J = F × t and also J = m × (v2 - v1). The two expressions always agree, which is why a problem can be solved from either end.

The unit falls straight out of the definition. A newton is a kilogram meter per second squared, so a newton-second is a kilogram meter per second, exactly the unit of momentum. Nothing has to be converted when you move between impulse and momentum, and that is why the two ideas are taught as one chapter.

Two ways into the same number

Every impulse question hands you one of two starting points. If you know the force and the contact time, multiply them. If you know the mass and the speed before and after, subtract the speeds and multiply by the mass. The calculator accepts either route and shows the other one as a cross-check, so an answer that disagrees with itself is easy to spot.

A worked case: a 2 kg ball leaves a boot at 25 m/s after arriving at 8 m/s. The speed change is 25 - 8 = 17 m/s, so the impulse is 2 × 17 = 34 newton-seconds. If the boot stayed in contact for 0.045 seconds, the average force was 34 / 0.045 = 755.6 newtons, which is 755.6 / 4.4482 = 169.9 pounds-force.

Run those numbers backwards and the arithmetic closes. A steady 755.6 newtons for 0.045 seconds is 34 newton-seconds, and 34 newton-seconds spread over a 2 kg ball is 17 m/s of speed change. One equation, three ways to read it.

Newton-seconds, pound-seconds and the units between them

The SI unit is the newton-second, written N·s. It is identical to the kilogram meter per second used for momentum, which makes the two quantities easy to compare. In US customary units the impulse unit is the pound-force second, written lbf·s, and the conversion is exact enough for any workshop: 1 lbf·s = 4.4482 N·s, so the 34 N·s kick above is 34 / 4.4482 = 7.64 lbf·s.

Mixing the two systems is the usual source of a wrong answer. A 2 lb ball is 0.907 kg (2 × 0.4536 = 0.9072), and the familiar 2.2 pounds per kilo is a round-off (2.2 × 0.4536 = 0.998, which is 0.2 percent light). Grams against kilograms is the same trap in a different place: a 145 g baseball is 0.145 kg, and entering 145 multiplies every answer by a thousand.

Same impulse, different contact time

An impulse does not care how the force is spread out. Doubling the contact time halves the average force, and cutting the contact time to a hundredth multiplies the force by a hundred. That single trade is behind airbags, crumple zones, helmets, bent knees and the give in a boxing glove.

A 70 kg driver in a 31 mph crash is moving at 14 m/s and carries 70 × 14 = 980 newton-seconds of momentum. Stopping in 0.08 seconds on a stiff belt alone means 980 / 0.08 = 12,250 newtons through the chest. Stretch the stop to 0.3 seconds with an airbag and a collapsing front end and the average force falls to 980 / 0.3 = 3,266.7 newtons. The impulse is identical; the load on the body is a quarter of what it was.

The same table explains a propped door on a hinge, a dropped phone in a case and a gymnast's landing mat. None of them reduce the momentum that has to be removed. They only buy time to remove it, and time is the denominator in the force.

Worked case: catching a 90 mph fastball

A regulation baseball is about 0.145 kg. A 90 mph pitch travels at 40.23 m/s (90 × 0.44704 = 40.2336), so the ball carries 0.145 × 40.2336 = 5.83 newton-seconds of momentum toward the plate. Catching it means removing all of that momentum, and the impulse the hand and mitt absorb is 5.83 N·s.

Stop the ball in 0.005 seconds against a bare palm and the average force is 5.83 / 0.005 = 1,166 newtons, close to the weight of a small motorcycle. Let a padded mitt give for 0.05 seconds and the same catch costs 5.83 / 0.05 = 116.6 newtons, about the weight of a heavy bag of groceries. Pull the hand back as the ball arrives and the contact time grows again, which is the oldest coaching cue in the game and a direct application of J = F × t.

The same arithmetic runs the other direction at the plate. A bat that reverses a 90 mph pitch to 105 mph out does not just stop the ball, it sends it back, and the impulse is built from the full change of velocity rather than from the pitch speed alone.

Impulse at the hobby bench: reading a rocket motor code

Model rocket motors carry a code such as C6-5, and each part of it is a number a hobbyist can use. The letter sets the total impulse band. The National Association of Rocketry guide to motor codes lists A as 1.26 to 2.50 newton-seconds, B as 2.51 to 5.00, C as 5.01 to 10.0 and D as 10.01 to 20.0, and each letter up holds about twice the impulse of the letter below it. The first digit is the average thrust in newtons and the last is the delay in seconds before the ejection charge fires.

A C6 tops out at 10 N·s, and 10 / 6 = 1.67, so a full-strength C6 burns for roughly 1.7 seconds at its labelled average thrust. The letter, not the digit, is the measure of total push, which is why a long-burning B4 can carry more total impulse than a short C3 that sounds louder on the pad.

Total impulse is the area under the thrust curve, so a motor with a spike at ignition and a long tail can sit in the same class as one with a flat push. Test stands print both the peak thrust and the total impulse for exactly that reason.

The area under a force curve is the impulse

Real impacts never hold a constant force. A bat, a foot or a bumper loads up, peaks and falls away, and the force at any instant is read off a curve. The impulse is the area under that curve, which is why a force-time graph answers the question faster than a list of numbers does.

A rectangle 0.045 seconds wide and 755.6 newtons tall covers 34 newton-seconds. A triangle with the same base and twice the peak, 1,511 newtons, covers the same 34 newton-seconds, because a triangle is half of the rectangle drawn around it (1,511 / 2 = 755.5). A curve that peaks at 2,000 newtons and tapers off can land on the same total for the same reason. Peak force tells you what a structure must survive; total impulse tells you what the object's motion will do.

This is also why a phone accelerometer placed on a package can report the impulse of a drop without ever recording the peak cleanly. The sensor samples acceleration over time, multiplies by the package mass, and adds up the slices. The sum is the impulse, whether or not the peak was captured.

Four mistakes that skew an impulse answer

Dropping the sign is the first. Impulse is a vector, and a ball caught and thrown back changes velocity by the sum of the two speeds rather than their difference. A 0.145 kg ball arriving at 40 m/s and leaving at 40 m/s the other way takes 0.145 × 80 = 11.6 newton-seconds, twice the impulse of a plain catch (2 × 5.8 = 11.6). A negative impulse is not an error; it means the push opposed the motion.

Using the peak force as if it were the average is the second. The average force is the impulse divided by the contact time, and on a sharp impact the peak can be several times the average. A sensor that reports the peak will overstate the impulse unless the whole curve is integrated.

Mixing pound-mass and pound-force is the third. A pound of mass is 0.4536 kg, while a pound of force is 4.4482 newtons. They share a name and nothing else, and a calculation that swaps them lands off by a factor of 9.8 without any obvious symptom.

Milliseconds against seconds is the fourth. A 20 ms contact is 0.02 s, not 20 s, and a force computed from the wrong one is off by a factor of a thousand: 5.83 / 0.02 = 291.5 newtons against 5.83 / 20 = 0.2915 newtons. Checking the size of the answer against everyday forces, a push is tens of newtons, a hard hit is thousands, catches most slips before it leaves the page.