UNIFORMLY ACCELERATED RECTILINEAR MOTION: CHARACTERISTICS, FORMULAS - PHYSICAL - 2025

The uniformly accelerated rectilinear motion is that which passes on a straight line and in which the moving body increases or decreases its speed at a constant rate. This rate is the magnitude that describes the rate at which the speed changes and is called acceleration.

In the case of uniformly accelerated or varied rectilinear motion (MRUV), constant acceleration is responsible for changing the magnitude of the velocity. In other types of motion, acceleration is also capable of changing the direction and sense of speed, or even just changing direction, as in uniform circular motion.

Figure 1. Accelerated movements are the most frequent. Source: Pixabay.

Since acceleration represents the change in velocity over time, its units in the International System are m / s ² (meters over seconds squared). Like speed, acceleration can be assigned a positive or negative sign, depending on whether the speed increases or decreases.

An acceleration of say +3 m / s ² means that for every second that passes, the speed of the mobile increases by 3 m / s. If at the beginning of the movement (at t = 0) the velocity of the mobile was +1 m / s, then after one second it will be 4 m / s and after 2 seconds it will be 7 m / s.

In uniformly varied rectilinear motion, variations in the speed that moving objects experience on a daily basis are taken into account. It is a more realistic model than the uniform rectilinear motion. Even so, it is still quite limited, as it restricts the mobile to travel only on a straight line.

characteristics

These are the main characteristics of uniformly accelerated rectilinear motion:

-The movement always runs along a straight line.

-The acceleration of the mobile is constant, both in magnitude and in direction and sense.

-The mobile speed increases (or decreases) linearly.

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

Figure 2. Graph of the acceleration versus time for a uniformly varied rectilinear motion. Source: Wikimedia Commons.

-The graph of the velocity v with respect to t is a straight line (in green in figure 3), whose slope is equal to the acceleration of the mobile. In the example the slope is positive.

Figure 3. Graph of velocity versus time for a uniformly varied rectilinear motion. Source: Wikimedia Commons.

-The cut with the vertical axis indicates the initial speed, in this case it is 0.4 m / s.

-Finally, the graph of position x versus time is the curve shown in red in figure 4, which is always a parabola.

Figure 4. Plot of position versus time for a uniformly varied rectilinear motion. Source: modified from Wikimedia Commons.

Distance traveled from the v vs. graph. t

By having the graph v vs. t, calculating the distance traveled by the mobile is very easy. The distance traveled is equal to the area under the line that is within the desired time interval.

In the example shown, suppose that you want to know the distance traveled by the mobile between 0 and 1 second. Using this graph, see Figure 5.

Figure 5. Graph to calculate the distance traveled by the mobile. Source: modified from Wikimedia Commons.

The distance sought is numerically equivalent to the area of ​​the trapezoid shaded in figure 3. The area of ​​the trapezoid is given by: (major base + minor base) x height / 2

It is also possible to divide the shaded area into a triangle and a rectangle, calculate the corresponding areas and add them. The distance traveled is positive, whether the particle is going to the right or to the left.

Formulas and equations

Both the average acceleration and the instantaneous acceleration have the same value in the MRUV, therefore:

-Acceleration: a = constant

When the acceleration is equal to 0, the motion is uniform rectilinear, since the speed would be constant in this case. The sign of a can be positive or negative.

Since the acceleration is the slope of the line v versus t, the equation v (t) is:

-Speed ​​as a function of time: v (t) = v _o + at

Where v _o is the value of the initial velocity of the mobile

-Position as a function of time: x (t) = x _or + v _or t + ½at ²

When you do not have time, but instead you have speeds and displacements, there is a very useful equation that is obtained by solving the time of v (t) = v _or + at and substituting it in the last equation. Is about:

Solved exercises

When solving a kinematics exercise, it is important to ensure that the situation is adapted to the model to be used. For example the equations of uniform rectilinear motion are not valid for accelerated motion.

And those of the accelerated movement are not valid for a circular or curvilinear type movement, for example. The first of these exercises solved below combines two mobiles with different movements. To solve it correctly, it is necessary to go to the appropriate movement model.

-Solved exercise 1

To find out the depth of a well, a child drops a coin and at the same time activates his timer, which stops just when he hears the coin hitting the water. Its reading was 2.5 seconds. Knowing that the speed of sound in air is 340 m / s, calculate the depth of the well.

Solution

Let h be the depth of the well. The coin travels this distance in free fall, a uniformly varied vertical motion, with initial velocity 0, as the coin is dropped, and constant downward acceleration equal to 9.8 m / s ². Take a time t _m in doing this.

Once the coin hits the water, the sound caused by the click travels up to the child's ear, who stops the stopwatch upon hearing it. There is no reason to believe that the speed of sound changes as it rises up the well, so the motion of sound is uniform rectilinear. The sound takes time t _s to reach the child.

Equation of motion for the coin:

Where x and a of the equation for the position given in the previous section have been replaced by h and g.

Equation of motion for sound:

This is the familiar equation distance = speed x time. With these two equations we have three unknowns: h, tm and ts. For the times there is a relationship, it is known that everything takes 2.5 seconds to happen, therefore:

Equating both equations:

Clearing one of the times and substituting:

This is a quadratic equation with two solutions: 2.416 and -71.8. The positive solution is chosen, which is the one that makes sense, since the time cannot be negative and in any case it must be less than 2.5 seconds. For this time it is obtained by substituting the depth of the well:

-Solved exercise 2

A car traveling at 90 km / h approaches a cross street with a traffic light. When it is 70 m away, the yellow light comes on, which lasts for 4 seconds. The distance between the traffic light and the next corner is 50 m.

The driver has these two options: a) brake at - 4 m / s ² or b) accelerate at + 2 m / s ². Which of the two options allows the driver to stop or cross the entire avenue before the light turns red?

Solution

The driver's starting position is x = 0 just when he sees the yellow light come on. It is important to convert the units properly: 90 km / h equals 25 m / s.

According to option a), in the 4 seconds that the yellow light lasts, the driver travels:

While the yellow light lasts, the driver travels like this:

x = 25.4 + ½.2.4 ² m = 116 m

But 116 m is less than the available distance to get to the next corner, which is 70 + 50 m = 120 m, so he cannot cross the entire street before the red light comes on. The recommended action is to brake and stay 2 meters from the traffic light.

Applications

People experience the effects of acceleration on a daily basis: when traveling by car or bus, as these continually need to brake and accelerate to adapt the speed to the obstacles on the road. Acceleration is also experienced when going up or down in an elevator.

Amusement parks are places where people pay to experience the effects of acceleration and have fun.

In nature, uniformly varied rectilinear motion is observed when an object is dropped freely, or when it is thrown vertically upward and waited for it to return to the ground. If air resistance is neglected, the acceleration value is that of gravity: 9.8 m / s2.

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