9 Linear Momentum and Collisions
9.5 Collisions in Multiple Dimensions
Learning Objectives
By the end of this section, you will be able to:
- Express momentum as a two-dimensional vector
- Write equations for momentum conservation in component form
- Calculate momentum in two dimensions, as a vector quantity
It is far more common for collisions to occur in two dimensions; that is, the angle between the initial velocity vectors is neither zero nor [latex]180^\circ[/latex]. Let’s see what complications arise from this.
The first idea we need is that momentum is a vector; like all vectors, it can be expressed as a sum of perpendicular components (usually, though not always, an x-component and a y-component, and a z-component if necessary). Thus, when we write down the statement of conservation of momentum for a problem, our momentum vectors can be, and usually will be, expressed in component form.
The second idea we need comes from the fact that momentum is related to force:
Expressing both the force and the momentum in component form,
Remember, these equations are simply Newton’s second law, in vector form and in component form. We know that Newton’s second law is true in each direction, independently of the others. It follows therefore (via Newton’s third law) that conservation of momentum is also true in each direction independently.
These two ideas motivate the solution to two-dimensional problems: We write down the expression for conservation of momentum twice: once in the x-direction and once in the y-direction.
This procedure is shown graphically in Figure.
We solve each of these two component equations independently to obtain the x– and y-components of the desired velocity vector:
(Here, m represents the total mass of the system.) Finally, combine these components using the Pythagorean theorem,
Problem-Solving Strategy: Conservation of Momentum in Two Dimensions
The method for solving a two-dimensional (or even three-dimensional) conservation of momentum problem is generally the same as the method for solving a one-dimensional problem, except that you have to conserve momentum in both (or all three) dimensions simultaneously:
- Identify a closed system.
- Write down the equation that represents conservation of momentum in the x-direction, and solve it for the desired quantity. If you are calculating a vector quantity (velocity, usually), this will give you the x-component of the vector.
- Write down the equation that represents conservation of momentum in the y-direction, and solve. This will give you the y-component of your vector quantity.
- Assuming you are calculating a vector quantity, use the Pythagorean theorem to calculate its magnitude, using the results of steps 3 and 4.
Example
Traffic Collision
A small car of mass 1200 kg traveling east at 60 km/hr collides at an intersection with a truck of mass 3000 kg that is traveling due north at 40 km/hr (Figure). The two vehicles are locked together. What is the velocity of the combined wr