UNIFORM RECTILINEAR MOVEMENT: CHARACTERISTICS, FORMULAS, EXERCISES - PHYSICAL - 2025

The uniform rectilinear motion or constant speed is that in which the particle moves along a straight line and with constant speed. In this way the mobile travels equal distances in equal times. For example, if in 1 second you travel 2 meters, after 2 seconds you will have traveled 4 meters and so on.

To make an accurate description of the movement, be it uniform rectilinear or any other, it is necessary to establish a reference point, also called the origin, with respect to which the mobile changes position.

Figure 1. A car traveling along a straight road at constant speed has uniform rectilinear motion. Source: Pixabay.

If the movement runs entirely along a straight line, it is also interesting to know in which direction the mobile runs along it.

On a horizontal line, it is possible that the mobile goes to the right or to the left. The distinction between the two situations is made by signs, the usual convention being the following: to the right I follow (+) and to the left I sign (-).

When the speed is constant, the mobile does not change its direction or its sense, and also the magnitude of its speed remains unchanged.

characteristics

The main characteristics of the uniform rectilinear motion (MRU) are the following:

-The movement always runs along a straight line.

-A mobile with MRU travels equal distances or spaces in equal times.

-The speed remains unchanged both in magnitude and in direction and sense.

-The MRU lacks acceleration (no change in speed).

-Since the velocity v remains constant at time t, the graph of its magnitude as a function of time is a straight line. In the example in figure 2, the line is colored green and the velocity value is read on the vertical axis, approximately +0.68 m / s.

Figure 2. Graph of velocity versus time for an MRU. Source: Wikimedia Commons.

-The graph of the x position with respect to time is a straight line, whose slope is equal to the speed of the mobile. If the line of the graph x vs t is horizontal, the mobile is at rest, if the slope is positive (graph of figure 3), the velocity is also positive.

Figure 3. Graph of the position as a function of time for a mobile with MRU that started from the origin. Source: Wikimedia Commons.

Distance traveled from the v vs. graph. t

Know the distance traveled by the mobile when the v vs. graph is available. t is very simple. The distance traveled is equal to the area under the line and within the desired time interval.

Suppose you want to know the distance traveled by the mobile of figure 2 in the interval between 0.5 and 1.5 seconds.

This area is that of the shaded rectangle in figure 4. It is calculated by finding the result of multiplying the base of the rectangle by its height, the values ​​of which are read from the graph.

Figure 4. The hatched area equals the distance traveled. Source: modified from Wikimedia Commons.

Distance is always a positive quantity, regardless of whether it is going to the right or to the left.

Formulas and equations

In the MRU the average speed and the instantaneous speed are always the same and since their value is the slope of the graph x vs t corresponding to a line, the corresponding equations as a function of time are the following:

-Position as a function of time: x (t) = x _o + vt

When v = 0 it means that the mobile is at rest. Rest is a particular case of movement.

-Acceleration as a function of time: a (t) = 0

In uniform rectilinear motion there are no changes in velocity, therefore the acceleration is zero.

Solved exercises

When solving an exercise, make sure that the situation corresponds to the model to be used. In particular, before using the MRU equations, it is necessary to make sure that they are applicable.

The following solved exercises are problems with two mobiles.

Resolved exercise 1

Two athletes approach each other with a constant speed of 4.50 m / s and 3.5 m / s respectively, initially being separated by a distance of 100 meters, as indicated in the figure.

If each one keeps its speed constant, find: a) How long do they take to meet? b) What will be the position of each one at that time?

Figure 5. Two runners move with constant speed towards each other. Source: self made.

Solution

The first thing is to indicate the origin of the coordinate system that will serve as a reference. The choice depends on the preference of the person who solves the problem.

Usually x = 0 is chosen right at the starting point of the mobiles, it can be in the left or right corridor, it can even be chosen in the middle of both.

a) We are going to choose x = 0 on the left runner or runner 1, therefore the initial position of this is x ₀₁ = 0 and for runner 2 it will be x ₀₂ = 100 m. Runner 1 moves from left to right with speed v ₁ = 4.50 m / while runner 2 moves from right to left with speed of -3.50 m / s.

Equation of motion for the first runner

Equation of motion for the second runner

As the time is the same for both t ₁ = t ₂ = t, when they meet the position of both will be the same, therefore x ₁ = x ₂. Matching:

It is an equation of the first degree for time, whose solution is t = 12.5 s.

b) Both runners are in the same position, therefore this is found by substituting the time obtained in the previous section in any of the position equations. For example, we can use that of broker 1:

The same result is obtained by substituting t = 12.5 s in the position equation for runner 2.

-Solved exercise 2

The hare challenges the tortoise to run a distance of 2.4 km and to be fair gives him a half hour head start. In the game, the turtle advances at a rate of 0.25 m / s, which is the maximum it can run. After 30 minutes the hare runs at 2 m / s and catches up with the tortoise quickly.

After continuing for 15 more minutes, she thinks she has time to take a nap and still win the race, but falls asleep for 111 minutes. When she wakes up she runs with all her might, but the turtle was already crossing the finish line. Find:

a) With what advantage does the turtle win?

b) The instant of time in which the hare overtakes the tortoise

c) The moment in which the tortoise overtakes the hare.

Solution to)

The race starts at t = 0. The position of the turtle: x _T = 0.25t

The hare movement has the following parts:

-Rest for the advantage it gave to the turtle: 0 <t <30 minutes:

-Race to catch up with the turtle and keep running a little after passing it; in total there are 15 minutes of movement.

-Sleep for 111 minutes (rest)

-Wake up too late (final sprint)

The duration of the run was: t = 2400 m / 0.25 m / s = 9600 s = 160 min. From this time we take 111 minutes from the nap and 30 minutes ahead, which makes 19 minutes (1140 seconds). It means you ran for 15 minutes before going to sleep and 4 minutes after waking up for the sprint.

At this time the hare covered the following distance:

d _L = 2 m / s. (15. 60 s) + 2 m / s (4. 60 s) = 1800 m + 480 m = 2280 m.

As the total distance was 2400 meters, subtracting both values ​​it turns out that the hare was 120 meters away from reaching the goal.

Solution b)

The position of the hare before falling asleep is x _L = 2 (t - 1800), considering the delay of 30 minutes = 1800 seconds. Equating x _{T and} x _L we find the time in which they are:

Solution c)

By the time the hare is overtaken by the tortoise, it is asleep 1800 meters from the start:

Applications

The MRU is the simplest motion imaginable and therefore the first to be studied in kinematics, but many complex motions can be described as a combination of this and other simple motions.

If a person leaves his house and drives until he reaches a long straight highway on which he travels at the same speed for a long time, his movement can be described globally as an MRU, without going into further detail.

Of course, the person needs to go around a few times before entering and exiting the highway, but by using this movement model the duration of the trip can be estimated knowing the approximate distance between the starting point and the arrival point.

In nature, light has a uniform rectilinear motion whose speed is 300,000 km / s. Likewise, the movement of sound in air can be assumed to be uniform rectilinear with a velocity of 340 m / s in many applications.

When analyzing other problems, for example the movement of charge carriers inside a conductor wire, the MRU approximation can also be used to give an idea of ​​what happens inside the conductor.

References

1. Bauer, W. 2011. Physics for Engineering and Sciences. Volume 1. Mc Graw Hill. 40-45.
2. Figueroa, D. Physics Series for Sciences and Engineering. Volume 3rd. Edition. Kinematics. 69-85.
3. Giancoli, D. Physics: Principles with Applications. 6 ^th. Ed Prentice Hall. 19-36.
4. Hewitt, Paul. 2012. Conceptual Physical Science. 5 ^th. Ed. Pearson. 14-18.
5. Kirkpatrick, L. 2007. Physics: A Look at the World. 6 ^ta Editing abbreviated. Cengage Learning. 15-19.
6. Wilson, J. 2011. Physics 10. Pearson Education. 116-119.