From the point of view of physics, absolute rest does not exist. Each body and the particles that make it up are in constant motion relative to each other. An important kinematic quantity characterizing the movement is speed. In this article, we give linear velocity formulas for various types of bodies moving in space.
What is linear speed?
This is a physical quantity that shows how much space a body travels in space per unit of time. As a rule, speed is denoted by the letter v¯, where the bar symbol indicates that it is a vector quantity. Measured speed in meters per second (m / s), kilometers per hour (km / h), miles per hour (mph) and other units, assuming the ratio of distance to time.
The velocity vector v¯ shows the direction of the real movement of the body. In this, it differs from the acceleration vector, which is directed towards the acting force, but not towards the body’s movement, although they can coincide.
Instant and medium speed
How to find the linear speed? The formula, according to the definition of magnitude, you can write the following:
v¯ = dl¯ / dt.
Where dl¯ is the body displacement vector during dt. This speed is called instantaneous because it is calculated over an extremely short period of time dt. Instantaneous speed is actually not stable and constantly changing. For example, suppose a car is moving along a road. At first glance, we can assume that at any given time its instantaneous speed will be constant, however, this is not so. Instantaneous speed fluctuates. If the car’s speedometer is sensitive enough, then it captures these vibrations.
The formula for the linear velocity of the average is no different from that for the instantaneous one, however, it is measured over a longer period of time Δt:
v¯ = Δl¯ / Δt, where Δt >> dt.
In the example with the car above, although the instantaneous speed is fluctuating, the average speed remains constant with a certain accuracy over the entire portion of the path Δl¯.
When solving problems, as a rule, they use an average speed. The instantaneous value makes sense only in the case of motion with acceleration.
Uniform movement in a straight line
This is an idealized type of motion, which assumes that the body moves along a straight line in space for some period of time. In this case, the speed of the body does not change. Denoting the distance traveled by the symbol l, we obtain the formula:
l = v * t.
Here v = const.
This type of movement was considered by philosophers of ancient Greece. They believed that for the movement of bodies it is necessary to apply some force, therefore, the natural state of all surrounding objects is peace. Only with the advent of the Renaissance, thanks to the work of Galileo and Newton, it was shown that if external forces do not affect the body, then the uniformity and straightness of its movement is not violated.
Speed when driving in a straight line with acceleration
When an external force appears, its action on the body leads to a change in the speed of the body. In dynamics, this situation is described by Newton’s second law:
F¯ = m * a¯.
If the action of the force F¯ occurs on an initially stationary body of mass m, then the formula for finding the linear velocity at any time t takes the form:
v¯ = a¯ * t.
In this case, both vector quantities are directed in the same direction. This formula can be used to describe the acceleration of a vehicle.
Now suppose that the car moved at a certain speed v 0 ¯, and then began to stop. In this case, the corresponding kinematic equation will take the form:
v¯ = v 0 ¯ + a¯ * t.
Since the velocity modulus | v¯ | the car will decrease over time, in scalar form this equality will be written like this:
v = v 0 - a * t.
In this case, the velocity and acceleration vectors are directed in opposite directions.
All linear velocity formulas given in this paragraph describe rectilinear motion with constant acceleration.
Body rotation
By rotation is meant the type of motion in which the trajectory of a moving body is a circle. Rotation can occur around an axis or around a fixed point. The rotation of the wheel, planets in their orbits, and athletes during figure skating competitions are all examples of this type of movement.
By analogy with linear displacement, the main formula for the dynamics of rotation is the following:
M = I * α.
Here M and I are the moments of force and inertia, respectively, α is the angular acceleration.
It is convenient to use angular velocity rather than linear to describe the rotation. It is defined as follows:
ω = θ / t.
Where θ is the angle by which the body turned in time t. With the recorded acceleration α, the velocity ω is related by the following equality:
ω = α * t.
Radians are used to measure all angular quantities.
Linear Speed Formula
It was noted above that rotation is conveniently described in angular characteristics. Nevertheless, in some cases, it is important to know what the linear velocity is. The formula for this case is given below:
v = ω * r.
Here r is the radius of the circle equal to the distance from any point on the trajectory of the body to the axis of rotation. The formula connecting linear and angular velocity is easy to obtain on your own. To do this, it is enough to consider how much circumferential distance the body will cover in a known time t.
The above expression can be used to calculate the linear velocities of cosmic bodies, for example, our Earth, revolving around the Sun.
Linear speed and centripetal acceleration
Speed is a vector quantity. This means that the body receives acceleration not only when the magnitude of v changes, but also when its direction changes. The latter situation is realized during rotation. The instantaneous body velocity vector is always tangent to the circle. If for equal intervals of time the body describes equal angles relative to the center of rotation, then such a movement is uniform from the point of view of the velocity modulus.
Deviation from rectilinear motion during rotation occurs due to the action of centripetal force, which causes centripetal acceleration. It is always directed perpendicular to speed; therefore, its module cannot be changed. The centripetal acceleration a c can be calculated by the formula:
a c = v 2 / r.
The absolute value of acceleration a c shows how large the centrifugal forces associated with the inertia of a rotating body are. A practical example is skidding during a sharp bend. Note that with decreasing radius a c grows more slowly than with increasing linear velocity.
The task of determining the linear velocity of our planet
Everyone understands that if a car moves at a speed of 100 km / h, then this figure is quite large in comparison with the speeds that people encounter in everyday life. It is interesting to compare this figure with the speed of the Earth's rotation in its orbit.
To assess this speed, we take the following data:
- radius of the orbit - 150 million km;
- the period of one revolution is 365 earth days.
To determine the required value, we use the formula of linear and angular velocity:
v = ω * r.
The value of ω through the period T is determined as follows:
ω = 2 * pi / T.
Then for v we arrive at the equality:
v = 2 * pi * r / T.
Substituting data from the conditions of the problem, we obtain a linear speed of 107.5 thousand km / h! This figure means that our Earth moves in outer space 1000 times faster than a car moves on the road. We do not feel this gigantic speed, since the Earth’s gravitational forces drag the atmosphere along so that it is at rest relative to the surface of the planet.