Equations of Uniformly Accelerated Motion with Step-by-Step Solutions
Uniformly Accelerated Motion
Uniformly Accelerated Motion refers to the type of motion in which an object's velocity changes at a constant rate. In simple words, when acceleration remains the same throughout the motion, it is called uniform acceleration. Such motion usually happens in a straight line, so the direction also remains constant.
Understanding this concept is essential for solving problems in physics, as it forms the foundation for analyzing real-world situations like objects falling under gravity, cars accelerating on a straight road, or balls rolling down smooth inclined planes.
What is Acceleration?
Acceleration is defined as the rate of change of velocity with respect to time. It tells us how quickly an object is speeding up or slowing down. If an object’s acceleration is constant, then the net change of velocity per second is always the same.
Acceleration can be positive (increasing velocity) or negative (decreasing velocity, also known as deceleration). If acceleration varies with time, we call it non-uniform acceleration.
Understanding Uniformly Accelerated Motion
Uniformly accelerated motion occurs in a straight line when an object is acted upon by constant acceleration. The velocity of the object changes equally in each equal time interval.
For example:
- A free-falling object under gravity (neglecting air resistance).
- A ball sliding down a frictionless slope.
- Applying brakes to a cycle on a straight road.
Behavior in Uniformly Accelerated Motion
Usually, the right or upward direction is considered positive, while left or downward is negative. If an object at rest gets positive acceleration, it begins to move in the positive direction.
If the object’s acceleration is opposite to its velocity, it will slow down, stop, and then move in the reverse direction. For instance, when you throw a ball upwards, gravity decelerates it until it stops momentarily, then accelerates it downwards.
Equations of Uniformly Accelerated Motion
There are five main equations used to solve uniformly accelerated motion problems. Selecting the appropriate one depends on what values are given and what needs to be found.
Let’s define the variables:
- a: Acceleration
- t: Time of acceleration
- v₀ or u: Initial velocity
- v or vf: Final velocity after time t
- Δx or s: Displacement in time t
| Equation | Description |
|---|---|
| 1. v = v₀ + at | Final velocity after time t |
| 2. Δx = v₀t + ½at² | Displacement after time t |
| 3. Δx = ½(v₀ + v)t | Displacement as average velocity × time |
| 4. Δx = vt – ½at² | Another form for displacement |
| 5. v² = v₀² + 2aΔx | Relation between velocity and displacement |
How to Choose the Right Equation
If you know initial velocity, acceleration, and time, use v = v₀ + at or Δx = v₀t + ½at².
If you know both velocities and want time or displacement, use Δx = ½(v₀ + v)t or v² = v₀² + 2aΔx.
Choosing the correct formula saves time when solving numerical problems and helps avoid mistakes.
Step-by-Step Solution Example
Example: A car starts from rest (v₀ = 0) and accelerates uniformly at 4 m/s² for 3 seconds. Find its final velocity and displacement.
Step 1: Write down what is given:
- Initial velocity (v₀) = 0 m/s
- Acceleration (a) = 4 m/s²
- Time (t) = 3 s
Step 3: To find displacement (Δx): Δx = v₀t + ½at² = 0 + ½×4×9 = 18 m
Positive vs Negative Acceleration
| Type | Description | Example |
|---|---|---|
| Positive Acceleration | Velocity increases with time. | Ball rolling down a hill. |
| Negative Acceleration | Velocity decreases with time (deceleration). | Braking bicycle to stop. |
Instantaneous Acceleration
Instantaneous acceleration is the acceleration at a specific moment. It is found by taking the rate of change of velocity as the time interval approaches zero.
Formula: ainst = dv/dt
This is especially useful when acceleration is not constant and helps analyze how objects behave at every instant.
To calculate it:
- Differentiate the velocity function with respect to time.
- For position x(t), first take dx/dt to get velocity, then differentiate again to get acceleration (a = d²x/dt²).
Solved Example: Instantaneous Acceleration
Given x(t) = 2t + 0.7t³, find instantaneous acceleration at t = 3 s.
- First, velocity v(t) = dx/dt = 2 + 2.1t²
- Next, acceleration a(t) = dv/dt = 4.2t
- At t = 3 s, a(3) = 4.2 × 3 = 12.6 m/s²
Uniform vs Non-Uniform Acceleration
| Uniform Acceleration | Non-Uniform Acceleration |
|---|---|
| Acceleration remains constant throughout motion. | Acceleration changes with time or position. |
| Equations of motion are directly applicable. | Requires calculus or variable analysis. |
| Ex: Free-fall without air resistance. | Ex: Driving through city traffic. |
Practice Questions
- A train starting from rest accelerates uniformly at 2 m/s² for 6 seconds. Find its final velocity and distance covered.
- An object moves with an initial velocity of 5 m/s and a constant acceleration of 3 m/s². Calculate its velocity after 4 seconds and its displacement in that time.
- Differentiate between uniform acceleration and non-uniform acceleration with one example each.
Further Learning on Vedantu
- Acceleration Explained
- Understanding Velocity-Time Graphs
- Kinematics Equations
- Speed vs Velocity
- Motion in a Straight Line
Summary
Uniformly accelerated motion helps us understand how objects move when acceleration is constant. Use the relevant equations based on what quantities the problem provides.
Mastery of these concepts allows you to solve real-life and exam-based physics problems efficiently and accurately.
FAQs on Uniformly Accelerated Motion (UAM): Complete Guide for Students
1. What is uniformly accelerated motion?
Uniformly accelerated motion is defined as the motion of an object in a straight line when it experiences a constant acceleration. In this motion, the velocity of the object changes by equal amounts in equal intervals of time. Examples include free fall under gravity, motion of a ball down a smooth incline, or a vehicle accelerating at a constant rate.
2. What are the equations of uniformly accelerated motion?
The three main equations for uniformly accelerated motion are:
1. v = u + at
2. s = ut + (1/2)at²
3. v² = u² + 2as
Where:
- v: final velocity
- u: initial velocity
- a: acceleration
- t: time
- s: displacement
These equations apply when acceleration is constant and the motion is along a straight line.
3. What is the formula for uniform acceleration?
The most common formula for uniform acceleration is:
a = (v - u) / t
Where:
- a: acceleration
- v: final velocity
- u: initial velocity
- t: time taken
This formula gives the acceleration when the change in velocity and time interval are known.
4. What is the difference between uniform and non-uniform acceleration?
Uniform acceleration occurs when the acceleration of a body is constant throughout its motion. Non-uniform acceleration means the acceleration changes during the motion.
Key Differences:
- Uniform acceleration: Same change in velocity per unit time; easy to use equations of motion.
- Non-uniform acceleration: Change in velocity per unit time varies; often needs calculus or graphical analysis.
5. What are some real-life examples of uniformly accelerated motion?
Common examples include:
- Object in free fall under gravity (ignoring air resistance)
- Car accelerating in a straight path at constant rate
- Rolling ball down a frictionless incline
- Bicycle slowing down uniformly with brakes
6. How do you identify if a motion is uniformly accelerated?
Uniformly accelerated motion can be identified if:
- The object moves in a straight line
- The acceleration (rate of change of velocity) remains constant throughout the motion
- Velocity changes by equal amounts in equal time intervals
7. What is instantaneous acceleration?
Instantaneous acceleration is the rate of change of velocity at a particular instant of time. It is calculated as the derivative of velocity with respect to time:
a = dv/dt
This represents the acceleration at a specific moment, even if the overall acceleration is varying.
8. What is the graphical representation of uniformly accelerated motion?
Graphical representations:
- Velocity-time graph: Straight line with constant (non-zero) slope; area under the line represents displacement.
- Displacement-time graph: Parabola opening upwards (for positive acceleration); shows increasing slope with time.
- Acceleration-time graph: Straight horizontal line showing constant acceleration.
9. What is positive and negative acceleration?
- Positive acceleration: Velocity increases with time.
- Negative acceleration (deceleration): Velocity decreases with time.
Example: When a car speeds up, it undergoes positive acceleration; when it applies brakes and slows down, it experiences negative acceleration.
10. Why are equations of uniformly accelerated motion important?
Equations of uniformly accelerated motion:
- Allow calculation of unknown variables such as displacement, time, velocity, and acceleration when three quantities are given
- Are widely used in solving Physics problems for competitive exams and board tests
- Form the foundation of kinematics and mechanics topics
11. How do you solve numerical problems based on UAM?
Steps:
- Write down given data (u, v, a, t, s)
- Select the suitable UAM equation
- Substitute known values, ensuring correct units
- Solve for the required unknown variable
- Check that your answer is physically logical and units are consistent
12. Can the equations of uniformly accelerated motion be used for vertical motion?
Yes, these equations apply to vertical motion such as free fall or objects thrown upwards, where acceleration is usually g = 9.8 m/s² (acceleration due to gravity). Remember to take upward or downward directions as positive or negative, as per the problem's reference frame.
