# Mechanics

calculus-based physics for scientists and engineers

## Impulse and Momentum in Two Dimensions

### Momentum and Impulse in Two Dimensions

Forces acting over time change the momentum of objects. Let's review what we learned in one dimension, because the physics hasn't changed.

The impulse is the integral of the force over a time interval. The total impulse, or the impulse of the net force on an object is equal to the change in momentum of the object. $$\int_{t_i}^{t_f}\vec{F}(t)\, dt = \Delta \vec{p} = \vec{p}_f - \vec{p}_i$$ If the force is constant in time, this simplifies to $$\vec{F}\Delta t = \Delta \vec{p}$$

The time-average of a force is the constant force that would give you the same impulse over the same time interval as the original force, and is given by: $$\vec{F}_{avg} = \frac{1}{\Delta t}\int_{t_i}^{t_f}\vec{F}(t)\, dt$$ and by substituting the definition of the impulse, you get $$\vec{F}_{avg}\Delta t = \Delta \vec{p}$$ regardless of the time characteristics of the original force. As one might have guessed, the time average of a constant force is itself.

Working in two dimensions is just a matter of working with vector relationships instead of scalar ones. Also, the physics in one perpendicular dimension doesn't affect the other. You can work with each dimension separately.