Now that you’ve learned about momentum, we can explore more complex situations where momentum changes and the conditions that cause momentum to change. In this post, we’ll explain the impulse equation, show you how to calculate impulse, and explore how impulse and momentum are related.

Impulse

Definition Of Impulse And Its Relation To Force

When a force is applied to an object, that force will impact the object’s motion. Impulse is a concept in physics we use to quantify the impact that force has on an object over a period of time. An everyday example of impulse is when a football player exerts a force on the quarterback to change his momentum. The impulse is equal to the product of the force the player exerts and the time they are in contact. 

How Are Impulse And Momentum Related?

Impulse and momentum are very closely related to each other. An impulse affects the object’s motion and thus causes the object’s momentum to change. How much the object’s momentum changes is exactly equal to the impulse the object experiences. 

How To Calculate Impulse

Equation for Impulse

…where:

• J is impulse
• F is force
• t is time

Impulse Units

The units for impulse come from those of force (newtons) and time (seconds). Since impulse is equal to the product of force and time, the units for impulse are \text{N}\cdot\text{s}. 

There is also an acceptable alternative unit for impulse. Since the amount of impulse that an object experiences is equal to its change in momentum, we can also use the units of momentum to measure impulse: \text{kg}\cdot\text{m/s}. 

Change In Momentum Formula

If you know the initial and final momentum of an object, you can calculate its change in momentum by finding the difference between those values. However, in many situations you may not directly know the object’s momentum and will instead be given the mass and velocity before and after the force is applied. As you learned before, momentum is the product of mass and velocity. Therefore, the change in momentum can be calculated as the product of mass and the change in velocity. 

…where:

• \Delta{p} is the change in momentum
• m is mass
• \Delta{v} is the change in velocity 

Impulse-Momentum Theorem 

Definition Of The Impulse-Momentum Theorem

The impulse-momentum theorem states that the impulse an object experiences is equal to the object’s change in momentum. This theorem is represented by the equation:

If we substitute the equations for impulse and change in momentum, this theorem can also be expressed as:

F\cdot t =m\Delta{v}

Significance Of The Impulse-Momentum Theorem

The impulse-momentum theorem is an important concept in physics because it provides a relationship between force, time, and motion. This is extremely helpful for analyzing collisions between objects and understanding how manipulating the variables affects the outcome of a collision. 

One of the most important applications is in the car safety features that we enjoy today. In car accidents, cars experience a massive change in momentum as they go from a high speed to a full stop. The force of such a collision can be enormous and potentially fatal. The theorem tells us that if we can increase the amount of time it takes for the passenger’s momentum to become zero, we can reduce the amount of force the passenger experiences. For example, when an airbag deploys in an accident, it reduces the force on the person by slowing them down more gradually than if they were to hit the much harder dashboard. By increasing the time it takes to stop the person, the airbag decreases the force the passenger experiences and saves lives.  

Examples: Crash Testing

Let’s consider the following example of a crash test dummy with and without an airbag.

Crash Without an Airbag

Let’s try the following example. A 60\text{ kg} crash test dummy is traveling in a car moving at 30\text{ m/s}. The car collides with a wall and comes to rest. Without an airbag, the crash test dummy hits the steering wheel and comes to rest in 0.05\text{ s}. How much force does the dummy experience in the collision?

First, we will calculate the change in momentum: 

\Delta{p}=m\Delta{v}

\Delta{p}=(60\text{ kg})(0\text{ m/s} -30\text{ m/s})

\Delta{p}=-1{,}800\text{ kg}\cdot\text{m/s}

Now applying the impulse-momentum theorem:

J=F\cdot t =\Delta{p}

F(0.05\text{ s})=(-1{,}800\text{ kg}\cdot\text{m/s})

Dividing by time produces:

F=\dfrac{-1{,}800\text{ kg}\cdot\text{m/s}}{0.05\text{ s}}

F=-36{,}000\text{ N}

This massive amount of force would almost certainly cause serious and potentially deadly injuries to a real passenger. 

Crash With an Airbag

Now the same crash test is conducted but with an airbag that brings the crash test dummy to rest in 0.3\text{ s}. With the airbag, how much force does the dummy experience? 

Here, the change in momentum is the same but the time is greater. We will apply the same method as before to calculate the force, but substitute the longer time of 0.3\text{ s}:

F=\dfrac{-1{,}800\text{ kg}\cdot\text{m/s}}{0.3\text{ s}}

F=-6{,}000\text{ N}

While that increase in time would be almost imperceptible to the human eye, the result is a significant reduction in the collision force. By harnessing the impulse-momentum theorem, engineers have designed safety features like airbags, seatbelts, and crumple zones that save thousands of lives by increasing the time and decreasing the collision force on car passengers. 

For more examples, check out this video from GPB Education.

Conclusion

In this post, we learned that impulse is the product of force and time and is equal to the change in momentum. We can apply the impulse-momentum theorem to analyze collisions and understand how increasing the time of impact decreases the force an object experiences. Like many concepts in physics, this theorem has important applications to everyday life.