It is far more common for collisions to occur in two dimensions; that is, the initial velocity vectors are neither parallel nor antiparallel to each other. Let's see what complications arise from this. The first idea 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 the statement of conservation of momentum is written for a...

The understanding of the concept of reference frames is essential to discuss relative motion in one or more dimensions. When we say that an object has a certain velocity, we must state the velocity with respect to a given reference frame. In most examples, this reference frame has been Earth. For instance, if a statement reads that a person is sitting in a train moving at 10 m/s east, then it implies that the person on the train is moving relative to the surface of Earth at this velocity,...

Relative velocity is the velocity of an object as observed from a particular reference frame, or the velocity of one reference frame with respect to another reference frame. The concept of relative velocity can be used to describe motion in two dimensions. Consider a particle P and two reference frames S and S′. The position of the origin of S′ as measured in S is , the position of P as measured in S′ is , and the position of P as measured in S is , which can be evaluated by utilizing...

When a flat plate is submerged in a fluid, the fluid exerts pressure on the plate. This pressure can lead to many different phenomena, including drag and buoyancy. To understand the behavior of the fluid over a flat plate of variable width, it is essential to analyze the distribution of the pressure exerted.
The pressure distribution on the plate can be calculated by determining the force that acts on a differential area strip of the plate. Thus, the magnitude of the force is equal to the...

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...

In geometry, measuring the direct distance between two points on a plane is essential in various practical and theoretical applications. Whether in navigation, engineering, or computer graphics, determining the shortest path between two locations involves using the distance formula. This formula is derived from the Pythagorean Theorem, which relates the lengths of the sides of a right triangle. On a coordinate plane, the horizontal and vertical distances between two points serve as the legs of...