Uniformly Accelerated Motion

Introduction

When a body moves in a plane or in a straight line, three characteristics – distance, velocity, and acceleration – are employed to characterize its motion. The term “distance” or “displacement” is self-evident. The rate of change of position is represented by velocity, whereas the rate of change of velocity is represented by acceleration. 

Displacement, velocity and acceleration, all of these three are vector quantities. There are two types of acceleration, namely, uniform and non-uniform. The value and direction of a homogeneous acceleration are both constant. It’s critical to understand the equations of motion that explain an object’s motion under uniform acceleration.

Acceleration

The rate of change of the velocity vector is defined as acceleration. The rate of acceleration might be constant or variable. In the event of constant acceleration, the ratio of net change of velocity to total time taken determines its value.

The SI unit for acceleration is m/s2, and the acceleration dimensions are [M0 L1T-2]. We’re all aware of the concept of uniform motion: an object is said to be in uniform motion if it travels the same distance in the same amount of time.

Uniform Acceleration

The term “uniform acceleration” refers to the acceleration that does not change over time. The rate of change of velocity remains constant in such instances. Because acceleration is a vector quantity, even in the case of constant acceleration, the direction of motion remains constant. Vector notations can be omitted because the body is traveling in a single direction with a fixed magnitude of acceleration.

Free-falling objects are one example of constant acceleration.

1. A frictionless slope with a ball rolling down it
2. The brakes have been applied to a bicycle

Variable Acceleration

If the velocity of a body changes by uneven amounts in equal intervals of time, it is said to be moving with variable acceleration.

Acceleration on Average

The average acceleration of a body moving with varying acceleration is defined as the ratio of the entire change in velocity of the body during a motion divided by the total time elapsed.

Average acceleration = (total change in velocity)/(total time taken) = v / t.

Instantaneous Acceleration is a term used to describe a sudden increase in speed.

The object’s acceleration at a given moment in time or at a specific position in motion.

The Uniformly Accelerated Motion along with a Straight Line Formula

The five quantities, displacement x, time is taken t, beginning velocity v0, end velocity v, and acceleration a, are connected by a set of simple equations termed kinematic equations of motion for objects in uniformly accelerated rectilinear motion:

• Relationship between velocity, acceleration and time: v = v0 + at
• Relationship between position, velocity and time: x = vo t + ½ at2
• Relationship between position, velocity, and acceleration: v2 ‒ vo2 = 2 a x

If the object’s location at time t = 0 is 0, In the preceding equations, x is replaced with

 (x – x0) if the particle starts at x = x0.

Uniformly Accelerated Motion Equations

Following a thorough knowledge of uniform acceleration, students should learn about the three kinematic equations that determine such a motion.

Keeping the value of acceleration constant in this scenario, the motion may be represented using equations. Assume that an object’s starting velocity was “u,” and that a constant force is applied, causing the body to move with constant acceleration “a,” and that the body reaches the velocity “t” while traveling the distance “s”.

Motion’s First Equation

Its value is provided by, in the case of constant acceleration,

v = u + at

Where v is the end velocity of the body, u is the beginning velocity of the body, a is the acceleration of the body, and t is the time interval.

Motion’s Second Equation

S = ut + (½) at2 is the distance formula.

Where u is the body’s starting velocity, a denotes the body’s acceleration, and t denotes the time interval.

Motion’s Third Equation

v2 = u2 + 2as ( velocity in terms of displacement)

Where v denotes the body’s ultimate velocity, u indicates the body’s starting velocity, a is the body accelerates and s denotes the total distance traveled.

Characteristics of Uniformly Accelerated Motion

1. Positive acceleration occurs when the direction of both acceleration and velocity change is the same.
2. Negative acceleration, also known as retardation or slowdown, occurs when the direction of acceleration and the change in velocity are opposite.
3. Positive acceleration causes the object’s speed to rise or decrease, whereas negative acceleration causes the object’s speed to slow down, which is known as retardation.

Non-uniform Acceleration

Uniform acceleration is the polar opposite of non-uniform acceleration. We know that uniform acceleration indicates that the rate of change in velocity will be consistent across time. 

Non-uniform acceleration occurs when an object’s velocity increases in uneven quantities at equal intervals of time. The magnitude of the acceleration and the direction of velocity will change over time.

Non-uniform acceleration can be seen in the following situations:

1. I) Obstacle rally in a busy street 
2. ii) Car travelling on a crowded street

Kinematic equations for uniformly accelerated motion. There are 3 kinematic equations of rectilinear motion for constant acceleration. Let’s understand them with the help of a few examples.

Problem 1

In 10 seconds, a youngster on a bicycle accelerates from 5 m/s to 20 m/s.

1. a) What is the bicycle’s rate of acceleration?
2. b) How far did the bicycle travel in the first ten seconds?

Solution: 

1. a) The beginning velocity u = 5 m/s and the end velocity v = 20 m/s in this issue. The bicycle’s acceleration is the rate of change of velocity and is calculated as follows:

a= (v – u)/ t= ( 20 m/s – 5 m/s)/10 seconds = 1.5 m/s2 

1. b) There are two methods for calculating the bicycle’s distance travelled in t = 10 seconds.
2. i) x = (1/2)(v + u) t = 0.5 (20 + 5) * 10 = 125 m 
3. ii) x = (1/2) at2 + u t = 0.5 * 1.5 * 100 + 5 * 10 = 125 m 

Problem 2

A 20 m/s item is flung straight down from the top of a skyscraper. It lands at a speed of 40 m/s.

1. a) What is the height of the structure?
2. b) For how long did the thing remain in the air?

Solution: 

The positive direction of the falling object is considered to be from the ground up. The initial (-20 m/s) and final (-40 m/s) velocities are provided; the minus sign was added to account for the fact that the falling item is travelling backwards. 

We know the gravitational acceleration acting on the falling object (g = – 9.8 m/s2), and we’re supposed to figure out how tall the structure is. If we take the object’s location to be x (with x = 0 on the ground), we may utilize the equation linking the initial and final velocities u and v, the acceleration a, and the initial (x0, the building’s height) and final (x, on the ground) positions as follows:

v2 = u2 + 2 a (x – x0)  

(-40 m/s)2 = (-20 m/s)2 + 2 (-9.8 m/s2) (0 – x0)

Solve the above for x0

x0 = 1200 / 19.6 = 61.2 m

1. b) x – x0 = (1/2)(u + v)t

-61.2 = 0.5(-20 – 40)t

t = 61.2 / 30 = 2.04 s

Conclusion

Uniformly Accelerated Motion (UAM) is the motion of an object with a constant acceleration. to estimate the acceleration due to gravity by measuring the location of an item in free fall as a function of time. To put it another way, the acceleration remains constant; it is equivalent to a number that does not vary as a function of time. 

Uniformly accelerated motion occurs when an object’s speed increases at a constant pace. As the distance traveled rises, so does the instantaneous velocity. The acceleration may be calculated using the constant of proportionality between the square of the velocity and the distance traversed.