Motion in a Straight Line Derivation, Formulas & Problems

In Physics, when the position of an object changes over a period of time is known as motion. Mathematically the motion is described in terms of displacement, distance, velocity, speed, acceleration, and time. By attaching the frame of reference, the motion of a body is observed. Further, based on the change in position of the body relative to the frame, the motion is measured.

What is Motion in a Straight Line?

The main aspects of motion in a straight line is discussed in this course by including the difference between the distance and displacement, average velocity and speed, acceleration along with the exercise of discussion. Further, solving the problem will grant students a holistic idea about the mechanics of motion in a straight line.

The student should accumulate the knowledge and skills through this designed-course. In each section, various ideas are explained in a simplified manner for the understanding of the student.

Types of Linear Motion

The two types of linear motion can be stated as follows:

Uniform linear motion

Non-Uniform linear motion

A body is known to be in uniform motion if it covers equal distance in equal motion time-span. Here, the motion is with zero acceleration and constant velocity.

Whereas, a body is known as non-uniform if it covers unequal distance in equal time-span. It comprises with non-zero acceleration and variable velocity

Equations of Motion Along a Straight Line

Calculus is the best way to derive the equation governing the motion in a straight line. If the value of the three relations velocity-time, distance-time and acceleration-time is known in the mathematical form, the value of the others can be obtained by differentiation or integration

Since

d/dt (distance) = velocity (v)

and

v/vt (velocity) = acceleration (a)

There is another method known as the graphical method, which can be used if a precise mathematical relation cannot be obtained. The below figure shows the graphical representation motion of a horse during a race and how the significant features of each graph are related to others.

Motion in a Straight Line Formulas

Constant Acceleration

This segment should be entitled "One-dimensional equations of motion for constant acceleration" for the sake of precision, as it will be a nightmare for a stylistic till let me begin this section with the following relations.

Velocity-time

During a uniform acceleration, the line of motion is straight; the longer the acceleration greater will be the change in velocity. Hence the relation between velocity and time will be simple during the uniform acceleration.

a= ∆v / ∆t

Enlarge ∆v to v − v₀ and condense ∆t to t.

a= (v−v₀) / t

Then resolve for v as a function of t.

v = v₀ + at

In the second equation of motion is written like a polynomial - a constant term (s0), followed by a first-order term (v0t), and followed by a second-order term (½at2). Since the maximum order is 2, it's more accurate to call it a quadratic.

∆s = v₀t + ½at²

The third equation of motion - In this once again, the symbol s0 is the initial stance, and s is the position some time t later. If you prefer, you may pen the equation using ∆s — the change in stance, displacement, or distance as the situation merits.

v² = v₀² + 2a∆s

Indeed, a quick solution, it wasn't that difficult compared to the first two derivations. It, however, worked because acceleration was constant in time and space.

Below are the formulas of motion in a straight line:

v =u + at

s=ut+1/2at²

v² = u² + 2as

Linear Motion Definition

A one-dimensional gesture along a straight line and which can be described by using only one longitudinal dimension is known as linear motion or rectilinear motion. The linear motion is basically divided into two types: one is uniform linear motion with constant velocity or zero acceleration, and the second one is non-uniform linear motion with a variety of velocity or non-zero acceleration. The movement of a particle along the line can be described by its position, which varies with time. For example, an athlete running 100m along a straight track is known as linear motion.

It is one of the most basic motions. As per Newton's first law of motion, any object that doesn't feel any net force will continue to go in a straight line with a perpetual velocity until it is subjected to a net force. In every-day circumstances, external forces such as friction and gravity can cause a change in the direction of its motion; hence its motion cannot be described as linear.

FAQ (Frequently Asked Questions)

1. Which Body is Considered Approximately a Point Object?

A railway carriage is running without any jolts between two railway stations.

A monkey is resting on top of a man who is cycling effortlessly on a circular track.

A spinning cricket ball that turns steeply on hitting the cricket pitch.

A tumbling beaker that has tumbled off the side of a table

Solution

An object can be regarded as a point object if the size of the object is much lesser than the distance it moves in a reasonable duration of time.

The size of a carriage is minor as related to the distance between two stations. Therefore, the carriage can be regarded as a point-sized object.

The size of a monkey is minimal as compared to the size of a circular track. Therefore, the monkey can be deemed as a point-sized thing on the trail.

The size of a rotating cricket ball is equivalent to the distance through which it turns sharply on hitting the cricket pitch. Hence, the cricket ball cannot be deemed as a point object.

The size of a beaker is equivalent to the height of the table from which it slipped. Hence, the beaker cannot be deemed as a point object. Thus (a) and (b) are correct Solutions

2. Which Proper Entries are in the Brackets Below?

The position-time (x-t) graphs for two children A and B coming back from their school O to their homes, P and Q respectively are shown.

(A/B) lives nearer to the school than (B/A)

(A/B) departs from the school earlier than (B/A)

(A/B) walks quicker than (B/A)

A and B get home at the (same/different) time

(A/B) surpasses (B/A) on the road (once/twice).

Solution

As OP < OQ, A lives nearer to the school than B

For x = 0, t = 0 for A; while t has some fixed value for B. Therefore, A head starts from the school earlier than B

We know that velocity is equivalent to the slope of the x-t graph. Now, since the slope of the x-t graph of B is better than that of A; therefore, B walks quicker than A.

It is evident from the given graph that both A and B reach their individual homes at the same time.

The x-t graph meets only once for A and B. Also, B moves later than A, and his/her speed is more than that of A. Hence B surpasses A only once on the road.