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How a Force vs. Time Graph Helps You Calculate Impulse, Momentum, and Velocity

airbag impulse momentum theorem example
Credit: Public Domain

By Andrew Bennett

The momentum of an object is the product of its mass and its velocity. A force must act on an object to change its momentum. The size of the force and the time over which that force acts both determine how much the momentum changes. In fact, the product of the force and the time are exactly equal to the change in momentum of the object. This product is called the "impulse delivered by the force."

Impulse-Momentum Theorem and Newton's Second Law

Newton's 2nd Law Definition

Newton's 2nd law says that the acceleration of an object (a) is equal to the force on it (F) divided by its mass (m), or:

Force vs. Time Graph Impulse Momentum Velocity.

The acceleration of the object can be calculated as the change in its velocity over the change in time, so we can replace acceleration in the equation above and get:

Force vs. Time Graph Impulse Momentum Velocity.


Multiplying both sides by mass and by time gives the result:

Force vs. Time Graph Impulse Momentum Velocity.

With the product of mass and velocity in that equation, it begins to look like a momentum calculation.

Impulse-Momentum Theorem Definition

The mass times the change in velocity of an object is the same as the change in momentum of that object, so we can rewrite the previous equation as:

Force vs. Time Graph Impulse Momentum Velocity.

The force on an object multiplied by the amount of time the force acts is equal to the change in momentum of the object. Impulse is defined as the force on an object multiplied by the time over which that force acts.

The Impulse-Momentum Theorem states that the impulse delivered to an object or system is equal to its change in momentum. One common case, which we'll address in the "conservation of momentum" post, is when there is no force acting on a system. In this case, the change in momentum will be zero, meaning that the total momentum of the system does not change.

Solving Impulse-Momentum Problems

Our first video on this topic further explains the relationship between Newton's 2nd Law and the Impulse-Momentum Theorem. We then apply the equation to an example problem involving a car applying its brakes to find its new velocity.


What Is a Force vs. Time Graph?

When the force is constant, we can calculate impulse by simply multiplying force by time. Yet in some cases, the force is changing. We can't just pick one value of the force to multiply by time. 

The solution is a very basic form of calculus. If you're not a fan of calculus, don't worry. You only need to find the area of triangles and rectangles.

When force changes over time, we can calculate impulse by graphing force vs. time. We will then calculate the area between our graphed values and the X-axis. Typically, this is called the "area under the curve," but the graph doesn't have to be curved. The area might actually be "over the curve" if we have negative values for force.

Using a Force vs. Time Graph to Find Velocity and Momentum 

After we find a value for impulse, we can calculate new values for momentum or velocity. We can do this because impulse is equal to the change in momentum. 

In our second video, we work through a force vs. time graph calculation to find impulse. Then, we use that value to calculate the change in momentum.


More Physics Videos

For more information on this topic, check out the other videos in this momentum playlist and subscribe to my YouTube channel.

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