Linear Speed Formula

The physical distance travelled by a moving item is always measured by linear speed. As a result, the linear speed is measured in path length units per unit of time. For instance, a meter per second. When an item moves in a circular motion, the term linear refers to smoothing out the curve that the object travels alongside the circle. It yields a line that is the same length. As a result, the standard definition of speed is correct: distance divided by time.

The distance between a point on a spinning object and the centre of rotation can be used to calculate its linear speed. The angular speed of an item is the angle it moves through in a given length of time. The angular speed will be expressed in radians per second (radian per second).

Given a complete circle, it has 2π radians. At a distance of r, or radius, from the rotation’s centre. The linear speed of a point on the object is thus equal to the angular speed multiplied by the distance r. Meters per second and meters per second is the unit of measurement.

V = ω × r

where,

• ω = speed in radians/ sec.
• r denotes the radius of the rotation

There are actually two common formulas for linear speed, depending on the situation:

For straight line motion:

The most basic formula for linear speed is:

v = d / t

where:

• v is the linear speed (often represented by v)
• d is the distance traveled (in meters, feet, etc.)
• t is the time taken to travel that distance (in seconds, hours, etc.)

This formula simply says that speed is equal to the distance traveled divided by the time it took to travel that distance.

For circular motion:

If you’re dealing with an object moving in a circle, you can find its linear speed using its angular speed and the radius of its circular path. The formula is:

v = ω × r

where:

• v is the linear speed (meters per second, feet per second, etc.)
• ω (omega) is the angular speed (radians per second)
• r is the radius of the circular path (meters, feet, etc.)

Angular speed describes how fast an object is rotating (measured in radians per second), and the radius tells you how far a point on the object is from the center of rotation. The linear speed relates these two to give you the actual speed of that point as it travels along the circle.

Question 1. Find the linear speed of a point on a wheel given that its speed is 14 RPS and diameter is 4 m.

Solution:

ω = 14 RPS or, 87.96 radians per second

 r = 4/2 = 2 m

Since, V = ω × r

= 87.96 × 2

V = 175.92 m/s

Question 2. Find the linear speed of a point on a wheel given that its speed is 8 RPS and diameter is 4 m.

Solution:

ω = 14 RPS or, 87.96 radians per second

r = 8/2 = 4 m

Since, V = ω × r

= 87.96 × 4

V = 351.84 m/s

Question 3. Find the linear speed of a point on a wheel given that its speed is 5 RPS and diameter is 2 m.

Solution:

ω = 5 RPS or, 31.42 radians per second

r = 2/2 = 1 m

Since, V = ω × r

= 31.42 × 1

V = 31.42 m/s

Question 4. Find the linear speed of a point on a wheel given that its speed is 19 RPS and diameter is 80 m.

Solution:

ω = 14 RPS or, 1.9897 radians per second

r = 80/2 = 40 m

Since, V = ω × r

= 1.9897 × 40

V = 79.588 m/s

Question 5. Find the linear speed of a point on a wheel given that its speed is 7 RPS and diameter is 1 m.

Solution:

ω = 7 RPS or, 87.96 radians per second

r = 1/2 = 0.5 m

Since, V = ω × r

= 43.98 × 0.5

V = 21.99 m/s