Rectilinear motion: definitions, time laws, and formulas

**Rectilinear motion** is the simplest of the various types of motion. Precisely for this reason it is one of the most requested physics topics by **high school students** who have to study or review for questions, class assignments and end-of-year exams, and by candidates for **admission tests to medicine, dentistry, veterinary medicine and health professions** .

In this article you will find everything you need to know about rectilinear motion: definitions, time laws, **formulas and exercises on uniform rectilinear motion and uniformly accelerated rectilinear motion** .

## Rectilinear motion: what is it

As we said from the beginning of this article, rectilinear motion is the simplest of the various cases of motion studied in physics. **Rectilinear motion develops along a straight line** , like the always straight road traveled by a motorbike. The straight line is the *trajectory* on which the movement occurs starting from a point called *the origin* .

To define rectilinear motion, **hourly laws have been developed. **The horary law describes the relationship between space and time and is used to calculate the amount of space covered by a moving body in a given period of time.

**The hourly law is an equation** denoted by *S(t)* in which:

- the independent variable is time, always with positive values;
- space is the dependent variable, which can have both positive and negative values depending on the direction of motion with respect to the origin.

The ratio between the distance traveled in a given period of time and the time interval is called **speed** :

*V = distance travelled/time taken = ΔS/Δt*

If the speed varies and does not remain constant, we speak of **acceleration** , which will be:

*a = speed/time variation = ΔV/Δt*

*Keep reading. *Now let’s delve into **two types of rectilinear motion** , whose definition and clockwise law we will explain in detail with formulas and examples:

- uniform rectilinear motion
- uniformly accelerated rectilinear motion

## Uniform rectilinear motion

Let’s immediately answer one of the most frequently asked questions among kids who study physics: **when is motion called uniform rectilinear motion? **Based on the **definition of uniform rectilinear motion** :

rectilinear motion is called uniform rectilinear motion when, given a body moving on a straight line, its speed is constant, i.e. it always travels the same amount of space in the same period of time.

Therefore, if a body moves with uniform rectilinear motions, there are no variations either in direction or direction, because the trajectory is a straight line, or in speed, because it is constant.

As we said in the previous general paragraph, the speed will therefore be the variation of space over time, measured in *m/s* (meters per second):

*V = distance travelled/time taken = ΔS/Δt*

in which:

*ΔS*is the variation of space,*S – S*, and is measured in meters,_{0}*m*.*Δt*is the change in time,*t – t*, and is measured in seconds,_{0}*s*.

From this **speed formula ΔS/Δt** , you can derive the **inverse speed formulas** :

*ΔS = V · Δt**Δt = ΔS/V*

Thus, the **hourly law of uniform rectilinear motion** is:

*S(t) = V · t + S _{0} ,* where:

- V is the speed, always constant
- t is the time
- S
_{0}is the starting position

### Exercise on uniform rectilinear motion

A motorbike travels along a straight road at a constant speed of 180 km/h.

*How many meters will he travel in 10 minutes?*

Read the problem carefully. The speed is always the same and the road is straight. Therefore the motion is uniform rectilinear.

The **problem data** are: *V = 180 km/h.*

The **unknowns to find** : the meters traveled in 10 minutes

So, since the speed is in km/h you will have to make an equivalence to convert km/h into m/s knowing that 1 km = 1000 m and in one hour there are 3600 s.

Don’t remember how to do equivalences? Read:Uniform Circular Motion

Continuing with our problem, we have that:

*180 km/h = (180 · 1000)/(1 · 3600) = 180000/3600 = 50 m/s*

Since there are 600 seconds in 10 minutes and given the inverse speed formula:

*ΔS = V · Δt = 50 · 600 = 30000 m = 30 km*

Our motorbike travels 30 km every 10 minutes.

## Uniformly accelerated rectilinear motion

The **definition of uniformly accelerated rectilinear motion** is: the motion of a body with constant acceleration along a straight trajectory always in the same direction and identical direction.

In practice, uniformly accelerated rectilinear motion is a **varied rectilinear motion** which, being characterized by constant acceleration, i.e. the speed always increases or decreases by the same amount, is called uniformly accelerated.

Therefore, the **formula for uniformly accelerated rectilinear motion** is:

*V(t) = a·t + V *_{0 ,} where:

- a is the acceleration, constant and expressed in m/s
^{2} - t is the time
- V
_{0}is the initial velocity

From this formula we can derive the **inverse formulas of uniformly accelerated rectilinear motions:**

*a = (V – V*_{0})/t = ΔV/t*t = (V – V*_{0})/a = ΔV/a*V*_{0}= V – a·t

Therefore, the **hourly law of uniformly accelerated rectilinear motions** is:

*S(t) = ½·a·t ^{2} + V _{0} ·t +* S

_{0}.

### Exercise on uniformly accelerated rectilinear motion

A motorbike starts from rest and travels 500 m in 20 seconds, constantly accelerating. *What is its acceleration and what is the final velocity reached after 20 seconds?*

The **problem data** is:

*V*, because the motorbike is stationary at the initial instant t_{0}= 0_{0}*ΔS = 500 m*, i.e. the space variation*Δt = 20 s*

The **unknowns to find** :

- a = acceleration
- V = speed

Given the hourly law of uniformly accelerated motion:

*ΔS = ½ a t ^{2} + V _{0} t +S _{0}* , with

*V*because the motorbike starts from rest and

_{0}= 0*S*because it starts from the origin

_{0}= 0*ΔS = ½·a·t ^{2} *and therefore

*a = 2ΔS/t*

^{2}Substituting the data, we have:

*a = 2 500/20 ^{2} = 1000/400 = 2.5 m/s ^{2}*

Regarding speed:

*V(t) = a·t +V _{0} = a·t = 2.5·20 = 50 m/s = 180 km/h.*

The acceleration of the motorbike is 2.5 m/s ^{2} and in 20 seconds, our motorbike reaches the speed of 180 km/h.

## Motion on an inclined plane

Finally, motion on an inclined plane. The **motions of a body on a smooth plane inclined** with respect to the horizontal axis by an assigned angle α is a uniformly accelerated rectilinear motions.

From the study of the forces that intervene on the motion of this body, it is clear that the movement does not depend on the mass but on the angle of inclination.

The acceleration on an inclined plane is less than that of gravity. The closer the inclination angle is to 90°, the closer the acceleration is to that of gravity.

Put simply, the more inclined the plane is, the faster the body will descend, a phenomenon that you will have noticed yourself in real life.

Therefore:

- the acceleration parallel to the plane is
*at*because the weight component is the only force along the horizontal axis;_{x}= gsin (α) - the acceleration perpendicular to the plane is
*at*because the body does not move along the vertical axis._{y}= 0,

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