WHAT IS LINEAR VELOCITY? (WITH SOLVED EXERCISES) - PHYSICAL - 2025

The linear velocity is defined as that which is always tangential to the path followed by the particle, regardless of shape is this. If the particle always moves in a rectilinear path, there is no problem in imagining how the velocity vector follows this straight line.

However, in general the movement is carried out on an arbitrarily shaped curve. Each portion of the curve can be modeled as if it were part of a circle of radius a, which at every point is tangent to the path followed.

Figure 1. Linear velocity in a mobile that describes a curvilinear path. Source: self made.

In this case, the linear speed is accompanying the curve tangentially and at all times at each point of it.

Mathematically the instantaneous linear velocity is the derivative of the position with respect to time. Let r be the position vector of the particle at an instant t, then the linear velocity is given by the expression:

v = r '(t) = d r / dt

This means that linear velocity or tangential velocity, as it is also often called, is nothing other than the change of position with respect to time.

Linear speed in circular motion

When the movement is on a circumference, we can go next to the particle at each point and see what happens in two very special directions: one of them is always pointing towards the center. This is the radial direction.

The other important direction is the one that passes on the circumference, this is the tangential direction and the linear speed always has it.

Figure 2. Uniform circular motion: the velocity vector changes direction and sense as the particle rotates, but its magnitude is the same. Source: Original by User: Brews_ohare, SVGed by User: Sjlegg.

In the case of uniform circular motion, it is important to realize that the velocity is not constant, since the vector changes its direction as the particle rotates, but its modulus (the size of the vector), which is the speed, yes it remains unchanged.

For this movement, the position as a function of time is given by s (t), where s is the arc traveled and t is time. In this case the instantaneous speed is given by the expression v = ds / dt and is constant.

If the magnitude of the speed also varies (we already know that the direction always does, otherwise the mobile could not turn), we are facing a varied circular movement, during which the mobile, in addition to turning, can brake or accelerate.

Linear velocity, angular velocity, and centripetal acceleration

The motion of the particle can also be seen from the point of view of the swept angle, rather than from the arc traveled. In this case we speak of the angular velocity. For a motion about a circle of radius R, there is a relationship between the arc (in radians) and the angle:

Deriving with respect to time on both sides:

Calling the derivative of θ with respect to t as angular velocity and denoting it with the Greek letter ω "omega", we have this relationship:

Centripetal acceleration

All circular motion has centripetal acceleration, which is always directed towards the center of the circumference. She ensures that the speed changes to move with the particle as it rotates.

The centripetal acceleration a _c or _R always points to the center (see figure 2) and is related to the linear velocity in this way:

a _c = v ² / R

And with the angular velocity as:

For a uniform circular motion, the position s (t) is of the form:

In addition, the varied circular motion must have an acceleration component called the tangential acceleration at _T, which is concerned with changing the magnitude of the linear velocity. If a _T is constant, the position is:

With v _o as the initial velocity.

Figure 3. Non-uniform circular motion. Source: Nonuniform_circular_motion.PNG: Brews oharederivative work: Jonas De Kooning.

Solved problems of linear velocity

The solved exercises help to clarify the proper use of the concepts and equations given above.

-Solved exercise 1

An insect moves on a semicircle of radius R = 2 m, starting from rest at point A while increasing its linear speed, at a rate of pm / s ². Find: a) After how long it reaches point B, b) The linear velocity vector at that instant, c) The acceleration vector at that instant.

Figure 4. An insect starts from A and reaches B on a semicircular path. It has linear speed. Source: self made.

Solution

a) The statement indicates that the tangential acceleration is constant and is equal to π m / s ², then it is valid to use the equation for uniformly varied motion:

With s _o = 0 and v _o = 0:

b) v (t) = v _or + to _T. t = 2π m / s

When at point B, the linear velocity vector points in the vertical direction down in the (- y) direction:

v (t) = 2π m / s (- y)

c) We already have the tangential acceleration, the centripetal acceleration is missing to have the velocity vector a:

a = a _c (- x) + a _T (- y) = 2π ² (- x) + π (- y) m / s ²

-Solved exercise 2

A particle rotates in a circle of radius 2.90 m. At a particular instant, its acceleration is 1.05 m / s ² in a direction such that it forms 32º with its direction of motion. Find its linear velocity at: a) This moment, b) 2 seconds later, assuming that the tangential acceleration is constant.

Solution

a) The direction of movement is precisely the tangential direction:

at _T = 1.05 m / s ². cos 32º = 0.89 m / s ²; a _C = 1.05 m / s ². sin 32º = 0.56 m / s ²

The velocity is solved from a _c = v ² / R as:

b) The following equation is valid for uniformly varied motion: v = v _o + a _T t = 1.27 + 0.89.2 ² m / s = 4.83 m / s

References

1. Bauer, W. 2011. Physics for Engineering and Sciences. Volume 1. Mc Graw Hill. 84-88.
2. Figueroa, D. Physics Series for Sciences and Engineering. Volume 3rd. Edition. Kinematics. 199-232.
3. Giancoli, D. 2006. Physics: Principles with Applications. 6 ^th.. Ed Prentice Hall. 62-64.
4. Relative Motion. Recovered from: courses.lumenlearning.com
5. Wilson, J. 2011. Physics 10. Pearson Education. 166-168.