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# instantaneous velocity calculator

From Wikipedia

Velocity

In physics, velocity is the measurement of the rate and direction of change in position of an object. It is a vectorphysical quantity; both magnitude and direction are required to define it. The scalarabsolute value (magnitude) of velocity is speed, a quantity that is measured in meters per second (m/s or msâˆ’1) when using the SI (metric) system.

For example, "5 meters per second" is a scalar and not a vector, whereas "5 meters per second east" is a vector. The average velocity v of an object moving through a displacement ( \Delta \mathbf{x}) during a time interval ( \Delta t) is described by the formula:

\mathbf{\bar{v}} = \frac{\Delta \mathbf{x}}{\Delta t}.

The rate of change of velocity is accelerationâ€“ how an object's speed or direction changes over time, and how it is changing at a particular point in time.

## Equation of motion

The velocity vector v of an object that has positions x(t) at time t and x(t + \Delta t) at time t + \Delta t, can be computed as the derivative of position:

\mathbf{v} = \lim_{\Delta t \to 0} \over \Delta t}={\mathrm{d}\mathbf{x} \over \mathrm{d}t}.

Average velocity magnitude is always smaller than or equal to average speed of a given particle. Instantaneous velocity is always tangential to trajectory. Slope of tangent of position or displacement time graph is instantaneous velocity and its slope of chord is average velocity.

The equation for an object's velocity can be obtained mathematically by evaluating the integral of the equation for its acceleration beginning from some initial period time t_0 to some point in time later t_n.

The final velocity v of an object which starts with velocity u and then accelerates at constant acceleration a for a period of time \Delta t is:

\mathbf{v} = \mathbf{u} + \mathbf{a} \Delta t.

The average velocity of an object undergoing constant acceleration is \tfrac {(\mathbf{u} + \mathbf{v})}{2}, where u is the initial velocity and v is the final velocity. To find the position, x, of such an accelerating object during a time interval, \Delta t, then:

\Delta \mathbf{x} = \frac {( \mathbf{u} + \mathbf{v} )}{2}\Delta t.

When only the object's initial velocity is known, the expression,

\Delta \mathbf{x} = \mathbf{u} \Delta t + \frac{1}{2}\mathbf{a} \Delta t^2,

can be used.

This can be expanded to give the position at any time t in the following way:

\mathbf{x}(t) = \mathbf{x}(0) + \Delta \mathbf{x} = \mathbf{x}(0) + \mathbf{u} \Delta t + \frac{1}{2}\mathbf{a} \Delta t^2,

These basic equations for final velocity and position can be combined to form an equation that is independent of time, also known as Torricelli's equation:

v^2 = u^2 + 2a\Delta x.\,

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.

In Newtonian mechanics, the kinetic energy (energy of motion), E_K, of a moving object is linear with both its mass and the square of its velocity:

E_{K} = \begin{matrix} \frac{1}{2} \end{matrix} mv^2.

The kinetic energy is a scalar quantity.

Escape velocityis the minimum velocity a body must have in order to escape from the gravitational field of the earth. To escape from the Earth's gravitational field an object must have greater kinetic energy than its gravitational potential energy. The value of the escape velocity from the Earth's surface is approximately 11100 m/s.

## 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 deal 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 vectorv 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:

\mathbf{v}_{A\text{ relative to }B} = \mathbf{v} - \mathbf{w}

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

\mathbf{v}_{B\text{ relative to }A} = \mathbf{w} - \mathbf{v}

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

### Scalar velocities

In the one dimensional case, the velocities are scalars and the equation is either:

\, v_{rel} = v - (-w), if the two objects are moving in opposite directions, or:
\, v_{rel} = v -(+w), 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), and an

Question:A train started from rest and moved with constant acceleration, At one time it was travelling at 30m/s and 160metres further on it was travelling at 50m/s, how do I find: 1. Acceleration 2.time required to travel the 160m mentioned

Answers:2. time required to travel the 160 m x = .5(vi + vf) * t x = 160 m vi = 30 m/s vf = 50 m/s t = ? 160 = .5(30 + 50) * t 160 = 40 * t t = 160 / 40 = 4 s 1. acceleration over the 160 m vf = vi + a * t 50 = 30 + a * 4 a = (50 - 30) / 4 = 5 m/s/s

Question:I am confused! I know instantaneous acceleration is tangent to acceleration graph. But then does that mean instantaneous velocity equal to acceleration since acceleration is tangent of velocity?

Answers:Instantaneous acceleration is the tangent of the **velocity** graph, not the tangent of the acceleration graph. That's probably why you're confused. If you have a graph of acceleration versus time, and you want to know the instantaneous acceleration at any particular time, just read the graph at that time - you don't need to measure the tangent or calculate the slope. You only need to use the tangent or slope if you need to figure out the acceleration from the graph of velocity versus time.

Question:If a particle has constant velocity, I understand how instantaneous velocity equals average velocity. That's easy to understand and visualize. The slope between two points would be the same as the slope of the tangent line, which is essentially the derivative of the line it self. But, to understand how this also applies to constant acceleration is hard for me. Constant acceleration means acceleration is increasing/decreasing at a constant rate. I would think instaneous acceleration would equal average acceleration, but I would not think instantaneous velocity would equal average velocity. I just don't understand! Please help!

Answers:Consider the case of a particle accelerating at a constant rate of 1 meter a second per second from rest. The instantaneous velocity at time 0 is zero meters per second. After 100 seconds, the instantaneous velocity is 100 meters per second, neither of which has any relationship with the average. Which is to say you appear to be right, but perhaps you - or I - have misunderstood the formulation. Maybe they intended to say or print the much more comprehensible "average acceleration", and not "average velocity".

Question:(a) Define: average velocity and speed, instantaneous velocity and acceleration. (b) Define the velocity and position of a projectile. A projectile has the least speed at what point in its path? Explain. (c) Describe uniform circular motion and determine the velocity and acceleration of an object which executes this motion. (d) A tiger leaps horizontally from a 16m high rock with a speed of 7.0 m/s. How far from the base of the rock will it land?

Answers:average velocity/speed refers to difference between the initial and the final and the average. instantaneious refers to the given speed at any given instant. A projectile has the least amount of speed once it has reached it's highest point which is ZERO before it accelerates back to the ground. d) t = square root of 2d/a t = square root of 2(16m) / 9.80m/s^2 t = 3.26s v = d /t d = v*t d = 7m/s * 3.26s = 22.9m