Understanding Uniformly Varied Motion: A Step-by-Step Guide

Oct 15, 2025 by ADMIN 60 views

Hey guys! Let's dive into the world of physics and tackle a classic problem involving uniformly varied motion. This type of motion is super common, and understanding it is key to grasping many other physics concepts. We'll break down a problem step-by-step, ensuring you grasp the core principles. This is a great way to learn, so let's get started. In our case, we're going to solve the problem of a mobile object, or a particle, experiencing uniformly varied motion. This is important because it involves acceleration, which is a fundamental concept in physics. So, buckle up, because we're about to explore how to calculate some of the key variables associated with this type of motion. This knowledge can be broadly applied, giving you a good basis for more complex calculations. Uniformly varied motion means that the acceleration is constant, which is a really helpful simplification in our calculations. Specifically, we'll be dealing with a mobile object that moves uniformly varied, starting from the origin of the Ox axis. So, let's go through the details step by step to help you fully understand the problem. It's important to note that uniformly varied motion is also known as uniformly accelerated motion. So, when you see the term "uniformly accelerated motion", it's the same thing as uniformly varied motion.

Problem Statement and Given Information

Understanding the Problem:

Let's first clarify the situation and what the problem is all about. It states that a mobile object starts its motion from the origin of the Ox axis. This means the initial position (x0) is zero. It begins with an initial velocity (v0) of 20 m/s at time t0 = 0 seconds. As the object moves, it passes through a point with an abscissa (position) of x = 150 meters at a certain time t. At this point, its velocity (v) is -10 m/s. Note that the negative sign here indicates that the velocity is in the opposite direction of the initial velocity.

What We Need to Find:

Now, the questions we need to answer are: a) What is the acceleration (a) of the mobile object? and b) At what moment of time (t) does the object pass through the point x = 150 m?

Given Data Summary:

Initial velocity, v0 = 20 m/s
Initial time, t0 = 0 s
Final position, x = 150 m
Final velocity, v = -10 m/s

This is the foundation of our analysis. Remember that the information given to us is crucial, as it provides the key variables that we can use to deduce the unknown.

Calculating the Acceleration (a) of the Mobile Object

Key Equations and Concepts:

To find the acceleration, we'll use a kinematic equation that relates initial velocity, final velocity, acceleration, and displacement, without the need for time. The key equation here is: v^2 = v0^2 + 2 * a * (x - x0). Where:

v is the final velocity
v0 is the initial velocity
a is the acceleration
x is the final position
x0 is the initial position

Since the object starts from the origin, x0 = 0. So, our equation simplifies to: v^2 = v0^2 + 2 * a * x.

Step-by-Step Calculation:

Rearrange the equation to solve for acceleration (a): a = (v^2 - v0^2) / (2 * x)
Plug in the given values: a = ((-10 m/s)^2 - (20 m/s)^2) / (2 * 150 m)
Calculate the values: a = (100 - 400) / 300 a = -300 / 300 a = -1 m/s^2

Result:

The acceleration of the mobile object is -1 m/s^2. The negative sign indicates that the acceleration is in the opposite direction to the initial velocity, meaning the object is decelerating. This makes sense because the object's velocity went from positive to negative, signifying the change in direction. We've successfully calculated the acceleration. Nice work, everyone!

Determining the Time (t) at Which the Object Passes Through x = 150 m

Key Equation and Concept:

Now that we know the acceleration, we can find the time it takes for the object to reach x = 150 m. We'll use another kinematic equation: v = v0 + a * t. Where:

v is the final velocity
v0 is the initial velocity
a is the acceleration
t is the time

Step-by-Step Calculation:

Rearrange the equation to solve for time (t): t = (v - v0) / a
Plug in the known values: t = (-10 m/s - 20 m/s) / (-1 m/s^2)
Calculate the time: t = -30 / -1 t = 30 s

Result:

The time at which the object passes through the point x = 150 m is 30 seconds.

Summary of Results and Interpretation

Key Findings:

Acceleration (a) = -1 m/s^2
Time (t) at x = 150 m = 30 s

Interpretation:

The object's acceleration is negative, indicating that the object is decelerating or slowing down. This is consistent with the change in velocity from positive to negative. The object takes 30 seconds to reach the point 150 m. This comprehensive analysis helps us understand how to solve these kinds of problems.

Practical Applications and Further Learning

Real-World Examples:

Understanding uniformly varied motion is essential in many real-world scenarios, such as:

Vehicle Motion: Analyzing the acceleration and deceleration of cars, trains, and other vehicles.
Projectile Motion: Calculating the trajectory of objects thrown or launched into the air, considering constant gravitational acceleration.
Sports: Analyzing the motion of athletes, like runners, swimmers, or divers.

Further Learning:

To deepen your understanding, explore:

Velocity-Time Graphs: Learn to visualize motion using graphs. The slope of a velocity-time graph represents acceleration.
Position-Time Graphs: Understand how the position of an object changes over time.
More Complex Problems: Try solving problems with varying acceleration or involving multiple phases of motion.

Keep practicing, and you'll master these concepts in no time. You're doing great, guys!

Conclusion and Key Takeaways

In conclusion, we've successfully solved a problem involving uniformly varied motion, calculating acceleration and time. We've seen how to use kinematic equations to analyze motion and have gained a deeper understanding of this fundamental concept. Remember to always start by identifying the given information, selecting the appropriate equations, and solving step-by-step. Practice is key! The more problems you solve, the more comfortable you'll become with these concepts. Always remember to check the units of your answers! This will help you catch any errors. Keep up the fantastic work. You got this!