9.12

Collisions in Multiple Dimensions: Problem Solving

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In a two-dimensional collision, momentum is conserved in the x and the y-direction independently.

For example, two skiers collide at an icy intersection and get stuck with each other. What will be the velocity of the skiers and their direction of travel after collision?

The two skiers form a closed system. In this case, as the skiers intersect at right angles, we assume the positive x-axis along the direction of the first skier and the positive y-axis along the direction of the second skier.

If the mass and velocity of the skiers before collision are known, then the velocity and direction of the skiers after collision can be resolved.

The equations for momentum conservation in the x and y direction are stated independently to resolve for the final momentum components.

The magnitude of the final momentum vector is obtained using the Pythagorean theorem from which the velocity of the skiers is obtained.

The direction of the skiers after the collision is obtained using the inverse tangent function for the momentum components.

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In multiple dimensions, the conservation of momentum applies in each direction independently. Hence, to solve collisions in multiple dimensions, we should write down the momentum conservation in each direction separately. To help understand collisions in multiple dimensions, consider an example.

A small car of mass 1,200 kg traveling east at 60 km/h collides at an intersection with a truck of mass 3,000 kg traveling due north at 40 km/h. The two vehicles are locked together. What is the velocity of the combined wreckage?

Firstly, a closed system is required. The natural system to choose is the car and truck, but this system is not closed, as friction from the road acts on both vehicles. We can avoid this problem by restricting the question to finding the velocity at the instant just after the collision so that friction has not yet affected the system; thereby, momentum is conserved for this system. Since there are two directions involved, we write equations for conservation of momentum twice: once in the x-direction and once in the y-direction.

Here, the known quantities are mass of the car (m_c= 1,200 kg), mass of the truck (m_t= 3,000 kg), magnitude of velocity of the car (v_c= 60 km/h) and magnitude of velocity of the truck (v_t= 40 km/h). The magnitude of the velocity of the wreckage v_wneeds to be calculated. The expression for the total momentum before the collision and after the collision is written down. Since the system is closed, the momentum must be conserved. Hence, the momentum before the collision is equal to the momentum after the collision.

Chemical equilibrium diagram with reaction rates; depicts forward, reverse reactions; dynamic balance.

Let us define the +x-direction as pointing east and the +y-direction as pointing north. Now, the momentums are resolved along the x and y directions.

Chromatography diagram: separation of mixture components, includes column setup for analysis.

Chemical structure of vanilla, C<sub>8</sub>H<sub>8</sub>O<sub>3</sub>, aromatic aldehyde compound diagram.

From these equations, the x and y components of the velocity of the wreckage are determined. Applying the Pythagorean theorem, the resultant velocity of the wreckage is calculated to be 33.3 km/h. The direction of velocity can be calculated using the expression,

Static equilibrium ΣF=0 diagram, physics concept, arrows illustrating forces and balance.

This gives the direction of velocity as 59°. This angle is towards the northeast, or 31° counterclockwise from the +x-direction.

This text is adapted from Openstax, University Physics Volume 1, Section 9.5: Collisions in Multiple Dimensions.

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