Linear MomentumEdit
Linear momentum is a foundational concept in classical mechanics that captures how much motion an object has and how resistant that motion is to changes. It is defined as the product of an object's mass and its velocity, written as p = m v. Because velocity is a vector, momentum is also a vector, pointing in the same direction as motion. The magnitude grows with both mass and speed. In everyday terms, momentum is a measure of how hard it is to stop a moving object or to change its direction. Key ideas about linear momentum are expressed through the relationships p = m v and the fundamental laws that connect force, motion, and time. For a deeper look at the quantity itself, see momentum and its relation to mass and velocity.
Momentum plays a central role in predicting and understanding motion across many systems. In Newtonian mechanics, the time rate of change of momentum equals the net external force acting on a body, F = dp/dt. This differential form of Newton's laws ties together motion and cause, and it underpins the idea that forces alter motion by changing momentum. The cumulative effect of force over a period of time is called impulse, J, with J = ∫ F dt, and for the same interval J = Δp, the change in momentum. See impulse for the detailed treatment of this concept and its applications.
Momentum is conserved in isolated systems—those free from external influences. In such systems, the total momentum remains constant, which is a powerful organizing principle in physics and engineering. This conservation is intimately connected to the translational symmetry of space and to the concept of a center of mass. The velocity of the center of mass is given by V_cm = P_total / M_total, where P_total is the total momentum and M_total the total mass. In a closed system, external forces are what can change the total momentum; otherwise, the sum of the momenta of all components stays fixed. See conservation of momentum and center of mass for more on these ideas.
Definition and vector form
Linear momentum is a vector quantity p = m v. The vector nature means both magnitude and direction matter, and momentum adds vectorially in systems of multiple bodies. In three dimensions, p = m v with v having components in the x, y, and z directions. The units of momentum are kilogram meters per second (kg·m/s). See vector and units for related fundamentals, and mass and velocity for the quantities that compose momentum.
Impulse and momentum change
The change in momentum of a body is caused by the external force acting on it over a time interval. If the net external force is F(t), then Δp = ∫ F dt, which is the impulse delivered to the body during the interval. This relationship is widely used in engineering to design collisions, impacts, and propulsion systems. See force (physics) and impulse for more details.
Conservation of momentum in collisions
In collisions, momentum is typically redistributed among the participating bodies, but the total momentum is conserved when external forces are negligible during the interaction. This principle explains a broad range of phenomena from billiards to particle physics. Elastic collisions conserve kinetic energy in addition to momentum, while inelastic collisions do not conserve kinetic energy, though momentum remains conserved. See elastic collision and inelastic collision for definitions and examples, and note how the coefficient of restitution e characterizes the elasticity of a collision.
Center of mass and reference frames
The center of mass provides a convenient way to analyze motion of a system. The total momentum equals the total mass times the velocity of the center of mass, P_total = M_total V_cm. In the absence of external forces, the center of mass continues with constant velocity, making it a natural frame for describing internal exchanges of momentum among parts of a system. See center of mass for a fuller treatment.
Relativistic momentum
At high speeds approaching the speed of light, the nonrelativistic expression p = m v ceases to be accurate. Special relativity modifies momentum to p = γ m v, where γ = 1 / sqrt(1 − v^2/c^2). As v approaches c, momentum grows without bound, reflecting the increasing inertia of rapidly moving bodies. Photons, despite having no rest mass, carry momentum p = E/c = h/λ, where E is energy, h is Planck's constant, and λ is wavelength. See relativistic momentum and special relativity for the full framework, and photon for momentum carried by light.
Applications and broader context
The concept of momentum is applied across physics and engineering to analyze propulsion, impact, and motion in fluids and solids. In aerospace and rocketry, momentum exchange with expelled propellant is the basis of thrust and propulsion efficiency, as captured in the rocket equation and related analyses. In sports and engineering design, momentum considerations help optimize outcomes in collisions or impacts. The conservation principle provides a robust predictive tool even when detailed forces are complex or uncertain, as long as external influence over the interaction interval can be neglected. See propulsion, collision, and center of mass for related discussions.
Historical context
The idea of momentum developed through the work of early scientists such as Isaac Newton and his contemporaries, who formalized laws of motion and the conservation principles that emerge from symmetry considerations. The evolution of these ideas spanned classical mechanics and, later, relativistic extensions, shaping how modern physics and engineering model motion, interactions, and energy transfer. See history of physics and conservation of momentum for more on how these concepts emerged and were refined.