Velocity

Velocity
As a change of direction occurs while the racing cars turn on the curved track, their velocity is not constant. Bạn đang xem: công thức vận tốc
Common symbols	v, v, v→, v
Other units	mph, ft/s
In SI base units	m/s
Dimension	L T⁻¹

Velocity is the tốc độ and the direction of motion of an object. Velocity is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of bodies.

Velocity is a physical vector quantity: both magnitude and direction are needed đồ sộ define it. The scalar absolute value (magnitude) of velocity is called speed, being a coherent derived unit whose quantity is measured in the SI (metric system) as metres per second (m/s or m⋅s⁻¹). For example, "5 metres per second" is a scalar, whereas "5 metres per second east" is a vector. If there is a change in tốc độ, direction or both, then the object is said đồ sộ be undergoing an acceleration.

Constant velocity vs acceleration

To have a constant velocity, an object must have a constant tốc độ in a constant direction. Constant direction constrains the object đồ sộ motion in a straight path thus, a constant velocity means motion in a straight line at a constant tốc độ.

For example, a siêu xe moving at a constant trăng tròn kilometres per hour in a circular path has a constant tốc độ, but does not have a constant velocity because its direction changes. Hence, the siêu xe is considered đồ sộ be undergoing an acceleration.

Difference between tốc độ and velocity

Kinematic quantities of a classical particle: mass m, position r, velocity v, acceleration a.

Speed, the scalar magnitude of a velocity vector, denotes only how fast an object is moving.^[1]^[2]

Equation of motion

Average velocity

Velocity is defined as the rate of change of position with respect đồ sộ time, which may also be referred đồ sộ as the instantaneous velocity đồ sộ emphasize the distinction from the average velocity. In some applications the average velocity of an object might be needed, that is đồ sộ say, the constant velocity that would provide the same resultant displacement as a variable velocity in the same time interval, v(t), over some time period Δt. Average velocity can be calculated as:

The average velocity is always less than vãn or equal đồ sộ the average tốc độ of an object. This can be seen by realizing that while distance is always strictly increasing, displacement can increase or decrease in magnitude as well as change direction.

In terms of a displacement-time (x vs. t) graph, the instantaneous velocity (or, simply, velocity) can be thought of as the slope of the tangent line đồ sộ the curve at any point, and the average velocity as the slope of the secant line between two points with t coordinates equal đồ sộ the boundaries of the time period for the average velocity.

The average velocity is the same as the velocity averaged over time – that is đồ sộ say, its time-weighted average, which may be calculated as the time integral of the velocity:

where we may identify

and

Special cases

When a particle moves with different uniform speeds v₁, v₂, v₃, ..., v_n in different time intervals t₁, t₂, t₃, ..., t_n respectively, then average tốc độ over the total time of journey is given as

If t₁ = t₂ = t₃ = ... = t, then average tốc độ is given by the arithmetic mean of the speeds

When a particle moves different distances s₁, s₂, s₃,..., s_n with speeds v₁, v₂, v₃,..., v_n respectively, then the average tốc độ of the particle over the total distance is given as

If s₁ = s₂ = s₃ = ... = s, then average tốc độ is given by the harmonic mean of the speeds

Instantaneous velocity

If we consider v as velocity and x as the displacement (change in position) vector, then we can express the (instantaneous) velocity of a particle or object, at any particular time t, as the derivative of the position with respect đồ sộ time:

From this derivative equation, in the one-dimensional case it can be seen that the area under a velocity vs. time (v vs. t graph) is the displacement, x. In calculus terms, the integral of the velocity function v(t) is the displacement function x(t). In the figure, this corresponds đồ sộ the yellow area under the curve labeled s (s being an alternative notation for displacement).

Since the derivative of the position with respect đồ sộ time gives the change in position (in metres) divided by the change in time (in seconds), velocity is measured in metres per second (m/s). Although the concept of an instantaneous velocity might at first seem counter-intuitive, it may be thought of as the velocity that the object would continue đồ sộ travel at if it stopped accelerating at that moment.

Relationship đồ sộ acceleration

Although velocity is defined as the rate of change of position, it is often common đồ sộ start with an expression for an object's acceleration. As seen by the three green tangent lines in the figure, an object's instantaneous acceleration at a point in time is the slope of the line tangent đồ sộ the curve of a v(t) graph at that point. In other words, acceleration is defined as the derivative of velocity with respect đồ sộ time:

From there, we can obtain an expression for velocity as the area under an a(t) acceleration vs. time graph. As above, this is done using the concept of the integral:

Constant acceleration

In the special case of constant acceleration, velocity can be studied using the suvat equations. By considering a as being equal đồ sộ some arbitrary constant vector, it is trivial đồ sộ show that

with v as the velocity at time t and u as the velocity at time t = 0. By combining this equation with the suvat equation x = ut + at²/2, it is possible đồ sộ relate the displacement and the average velocity by

It is also possible đồ sộ derive an expression for the velocity independent of time, known as the Torricelli equation, as follows:

Xem thêm: Góc mẹo vặt hữu ích: Giày Vans bị phai màu phải làm sao?

where v = |v| etc.

The above equations are valid for both Newtonian mechanics and special relativity. Where Newtonian mechanics and special relativity differ is in how different observers would describe the same situation. In particular, in Newtonian mechanics, all observers agree on the value of t and the transformation rules for position create a situation in which all non-accelerating observers would describe the acceleration of an object with the same values. Neither is true for special relativity. In other words, only relative velocity can be calculated.

Quantities that are dependent on velocity

The kinetic energy of a moving object is dependent on its velocity and is given by the equation

ignoring special relativity, where E_k is the kinetic energy and m is the mass. Kinetic energy is a scalar quantity as it depends on the square of the velocity, however a related quantity, momentum, is a vector and defined by

In special relativity, the dimensionless Lorentz factor appears frequently, and is given by

where γ is the Lorentz factor and c is the tốc độ of light.

Escape velocity is the minimum tốc độ a ballistic object needs đồ sộ escape from a massive toàn thân such as Earth. It represents the kinetic energy that, when added đồ sộ the object's gravitational potential energy (which is always negative), is equal đồ sộ zero. The general formula for the escape velocity of an object at a distance r from the center of a planet with mass M is

where G is the gravitational constant and g is the gravitational acceleration. The escape velocity from Earth's surface is about 11 200 m/s, and is irrespective of the direction of the object. This makes "escape velocity" somewhat of a misnomer, as the more correct term would be "escape speed": any object attaining a velocity of that magnitude, irrespective of atmosphere, will leave the vicinity of the base toàn thân as long as it does not intersect with something in its path.

Relative velocity

Relative velocity is a measurement of velocity between two objects as determined in a single coordinate system. Relative velocity is fundamental in both classical and modern physics, since many systems in physics khuyến mãi with the relative motion of two or more particles. In Newtonian mechanics, the relative velocity is independent of the chosen inertial reference frame. This is not the case anymore with special relativity in which velocities depend on the choice of reference frame.

If an object A is moving with velocity vector v and an object B with velocity vector w, then the velocity of object A relative to object B is defined as the difference of the two velocity vectors:

Similarly, the relative velocity of object B moving with velocity w, relative đồ sộ object A moving with velocity v is:

Usually, the inertial frame chosen is that in which the latter of the two mentioned objects is in rest.

Scalar velocities

In the one-dimensional case,^[3] the velocities are scalars and the equation is either:

if the two objects are moving in opposite directions, or:

if the two objects are moving in the same direction.

Polar coordinates

In polar coordinates, a two-dimensional velocity is described by a radial velocity, defined as the component of velocity away from or toward the origin (also known as "velocity made good"^{[citation needed]}), and a transverse velocity, perpendicular đồ sộ the radial one. Both arise from angular velocity, which is the rate of rotation about the origin (with positive quantities representing counter-clockwise rotation and negative quantities representing clockwise rotation, in a right-handed coordinate system).

The radial and traverse velocities can be derived from the Cartesian velocity and displacement vectors by decomposing the velocity vector into radial and transverse components. The transverse velocity is the component of velocity along a circle centered at the origin.

where

The radial speed (or magnitude of the radial velocity) is the dot product of the velocity vector and the unit vector in the radial direction.

where is position and is the radial direction.

The transverse tốc độ (or magnitude of the transverse velocity) is the magnitude of the cross product of the unit vector in the radial direction and the velocity vector. It is also the dot product of velocity and transverse direction, or the product of the angular tốc độ and the radius (the magnitude of the position).

such that

Angular momentum in scalar size is the mass times the distance đồ sộ the origin times the transverse velocity, or equivalently, the mass times the distance squared times the angular tốc độ. The sign convention for angular momentum is the same as that for angular velocity.

where

The expression is known as moment of inertia. If forces are in the radial direction only with an inverse square dependence, as in the case of a gravitational orbit, angular momentum is constant, and transverse tốc độ is inversely proportional đồ sộ the distance, angular tốc độ is inversely proportional đồ sộ the distance squared, and the rate at which area is swept out is constant. These relations are known as Kepler's laws of planetary motion.