Impulse (Physics): Definition, Equation, Calculation (W/ Examples)

Impulse is something of a forgotten character in the scientific stage production that is classical mechanics. In physical science, there is a certain practiced choreography at play in terms of the rules governing movement. This has given rise to the various ​conservation laws​ of physical science.

Think of impulse for now as "the real-life forcefulness of a given force." (That language will make sense soon!) ​It is a concept critical to understanding how to actively reduce the force experienced by an object in a collision.​

In a world dominated by large objects carrying humans at high speeds at all hours, it's a good idea to have a large contingent of the world's engineers working to help make vehicles (and other moving machines) safer using the basic principles of physics.

Impulse, mathematically, is the product of average force and time, and it is equivalent to change in momentum.

The implications and derivation of the impulse-momentum theorem are provided here, along with a number of examples illustrating the importance of being able to manipulate the time component of the equation to change the level of force experienced by an object in the system in question.

Engineering applications are continually being refined and designed around the relationship between force and time in an impact.

As such, impulse principles have played a role in, or at least helped explain, many modern safety features. These include seatbelts and car seats, the ability of tall buildings to "give" slightly with the wind, and why a boxer or fighter who rolls with a punch (that is, dips in the same direction the opponent's fist or foot is moving) sustains less damage than one who stands rigid.

•It is interesting to consider the relative obscurity of the term "impulse" as it is used in physics, not just for the aforementioned practical reasons but also because of the familiarity of the properties to which impulse is most closely related. Position (x or y, usually), velocity (the rate of change of position), acceleration (the rate of change of velocity) and net force (acceleration times mass) are familiar ideas even to lay people, as is linear momentum (mass times velocity). Yet impulse (force times time, roughly) is not.

Impulse (​J​) is defined as the change in total momentum ​p​ ("delta p," written ∆​p​) of an object from the established start of a problem (time ​t​ = 0) to a specified time ​t​.

Systems can have many colliding objects at a time, each with their own individual masses, velocities and momenta. However, this definition of impulse is often used to calculate the force experienced by a single object during a collision. A key here is that the time used is the ​time of collision​, or how long the colliding objects are actually in contact with each other.

Remember that the momentum of an object is its mass times its velocity. When a car slows down, its mass (probably) doesn't change, but its velocity does, so you would measure the impulse here ​strictly over the period of time when the car is changing​ from its initial velocity to its final velocity.

By rearranging some basic equations, it can be demonstrated that for a constant force ​F​, the change in momentum ∆​p​ that results from that force, or m∆​v​ = m(​**v_f – v_i​), is also equal to ​F**​∆t ("F delta t"), or the force multiplied by the time interval during which it acts.

•Units for impulse here are thus newton-seconds ("force-time"), just as with momentum, as the math requires. This is not a standard unit, and as there are no SI units of impulse, the quantity is often expressed instead in its base units, kg⋅m/s.

Most forces, for better or for worse, are not constant for the duration of a problem; a small force may become a large force or conversely. This changes the equation to J = ​**F_net**​∆t. Finding this value requires using calculus to integrate the force over the time interval ​t​:

\(J = \int F(t)\,dt\)

All of this leads to the ​impulse-momentum theorem​:

The theorem follows from Newton's second law (more on this below), which can be written F_net = ma. It follows from this that F_net∆t = ma∆t (by multiplying each side of the equation by ∆t). From this, substituting a = (v_f – v_i)/∆t, you get [m(v_f – v_i)/∆t]∆t. This reduces to m(v_f – v_i), which is change in momentum ∆p.

T, his equation, however, only works for constant forces (that is, when acceleration is constant for situations in which mass does not change). For a non-constant force, which is most of them in engineering applications, an integral is required to evaluate its effects over the time frame of interest, but the result is the same as in the constant-force case even if the mathematical path to this result is not:

\(\int Fdt = \int (ma)dt = \int m(dv/dt)dt = \int m
dv = m∆v = ∆p\)

You can imagine a given "type" of collision that can be repeated countless times – the slowing of an object of mass m from a given known velocity v to zero. This represents a fixed quantity for objects with constant mass, and the experiment could be run a number of times (as in car crash testing). The quantity can be represented by m​∆v.​

From the impulse-momentum theorem, you know that this quantity is equal to ​F​_net∆t for a given physical situation. Since the product is fixed but the variables ​F​_net and ∆t are free to individually vary, you can compel the force to a lower value by finding a means of extending t, in this case the duration of collision event.

Put a little differently, impulse is fixed given specific mass and velocity values. That means that whenever ​F​ is increased, ​t​ must decrease by a proportional amount and conversely. Therefore, by increasing the time of a collision, the force must be reduced; impulse cannot change unless ​something else​ about the collision changes.

•Ergo, this a key concept: shorter collision times = larger force = more potential damage to objects (including people), and vice versa. This concept is captured by the impulse-momentum theorem.

This is the essence of the physics underlying safety devices such as airbags and seatbelts, which increase the time it takes a human body to change its momentum from some velocity to (usually) zero. This diminishes the force the body experiences.

Even if the time is reduced by only microseconds, a difference that human minds can't observe, dragging out how long a person slows down by putting them in contact with an airbag for much longer than a short hit to the dashboard can dramatically reduce the forces felt on that body.

Impulse and momentum have the same units, so aren't they sort of the same thing? This is almost like comparing heat energy to potential energy; there is no intuitive way to manage the idea, only math. But generally, you can think of momentum as a steady-state concept, like the momentum you have walking at 2 m/s.

Imagine your momentum changing because you bump into someone who is walking slightly slower than you in the same direction. Now imagine someone running into you head-on at 5 m/s. ​The physical implications of the difference between merely "having" momentum and experiencing different changes in momentum are enormous.​

Until the 1960s, athletes who participated in the high jump – which involves clearing a thin horizontal bar about 10 feet wide – usually landed in a sawdust pit. Once a mat was made available, jumping techniques became more daring, because athletes could land safely on their backs.

The world record in the high jump is just over 8 feet (2.44 m). Using the free-fall equation ​**v_f²​ = 2​a**​d with a = 9.8 m/s² and d = 2.44 m, you find that an object is falling at 6.92 m/s when it hits the ground from this height – a little over 15 miles an hour.

What is the force experienced by a 70-kg (154-lb) high jumper who falls from this height and stops in a time of 0.01 seconds? What if the time is increased to 0.75 seconds?

\(J=m\Delta v=(70)(6.92-0)=484.4\text{ kgm/s}\)

For t = 0.01 (no mat, ground only):

\(F=\frac{J}{\Delta t}=\frac{484.4}{0.01}=48,440\text{ N}\)

For t = 0.75 (mat, "squishy" landing):

\(F=\frac{J}{\Delta t}=\frac{484.4}{0.75}=646\text{ N}\)

The jumper landing on the mat experiences ​less than 1.5 percent of the force​ that the uncushioned version of himself does.

Any study of concepts such as impulse, momentum, inertia and even mass should begin by touching at least briefly on the basic laws of motion determined by the 17th- and 18th-century scientist Isaac Newton. Newton offered a precise mathematical framework for describing and predicting the behavior of moving objects, and his laws and equations not only opened doors in his day but remain valid today except for relativistic particles.

​Newton's first law of motion​, the ​law of inertia​, states that an object with a constant velocity (including ​v​ = 0) remains in that state of motion unless acted on by an external force. One implication is that no force is required to keep an object moving regardless of the velocity; force is needed only to change its velocity.

​Newton's second law of motion​ states that forces act to accelerate objects with mass. When the net force in a system is zero, a number of intriguing properties of motion follow. Mathematically, this law is expressed ​F​ = m​a​.

​Newton's third law of motion​ states that for every force ​F​ that exists, a force equal in magnitude and opposite in direction (​–F​) also exists. You can probably intuit that this has interesting implications when it comes to the accounting side of physical science equations.

If a system does not interact with the external environment at all, then certain properties related to its motion do not change from the beginning of any defined time interval to the end of that time interval. This means that they are ​conserved​. Nothing disappears or literally appears from nowhere; if it is a conserved property, it must have existed previously or will continue to exist "forever."

Mass, momentum (two types) and ​energy​ are the most famously conserved properties in physical science.

•​Conservation of momentum:​ Adding up the sum of the momenta of the particles in a closed system at any instant always reveals the same result, regardless if the individual directions and speeds of the objects.
•​Conservation of angular momentum​: The angular momentum ​L​ of a rotating object is found using the equation m​vr​, where ​r​ is the the vector from of the axis of rotation to the object.
•​Conservation of mass:​ Discovered in the late 1700s by Antoine Lavoisier, this is often informally phrased, "Matter can neither be created nor destroyed."
•​Conservation of energy:​ This can be written in a number of ways, but typically, it resembled KE (kinetic energy) + PE (potential energy) = U (total energy) = a constant.

Linear momentum and angular momentum are both conserved even though the mathematical steps required to prove each law are different, because different variables are used for analogous properties.