Calculating Distance And Displacement Of A Mobile Object

Hey guys! Let's dive into a fun physics problem. We're going to figure out how far a mobile object travels when it reaches a final velocity, given a specific acceleration period. This stuff is super important for understanding how things move in the real world, from cars to rockets! We'll break it down step-by-step so it's easy to follow. Get ready to flex those math muscles!

Understanding the Problem: The Mobile Object's Journey

Okay, so the scenario is this: We have a mobile object (think of it as anything that can move, like a car, a ball, or even a tiny particle) that accelerates for a certain amount of time. We know a few key things: the final speed (or velocity) the object achieves and the duration of its acceleration. Our mission? To calculate the total distance the object covers during that time. This might sound a bit tricky, but don't worry, we'll use some handy physics formulas to crack this code. Think of it like a treasure hunt; we have clues (the known values), and we need to find the treasure (the distance).

To make sure we're all on the same page, let's nail down some of the basics. Acceleration is the rate at which an object's velocity changes. Velocity, in turn, is the speed of something in a specific direction. Since we are told that the object has a final velocity of 30 m/s, it means the object is moving at 30 meters every second. Also, the time that the object is accelerating is 120 seconds. This problem also deals with displacement, which is a vector quantity that refers to an object's change in position. In our case, it represents the total distance that the object will move. The most important thing to keep in mind is the concepts of the initial velocity of the object, the acceleration, the final velocity, and the time. This problem assumes that the initial velocity is 0 m/s because it does not state anything otherwise. However, with the appropriate changes, the problem can be solved even if the initial velocity is not 0 m/s. It is also important to note that acceleration is the change in velocity with respect to time. It is a vector quantity, which has magnitude and direction.

So, what's the plan? We'll need to figure out what equations apply to the situation. We can utilize kinematic equations to deal with this problem. Because acceleration is constant, and we have enough information, we can solve for the distance.

We know that the final velocity is 30 m/s. We know that the time is 120 seconds, and we are looking for the distance (displacement). The object starts with an initial velocity of 0 m/s. Because the mobile object's acceleration is constant, we can apply a couple of kinematic equations. First, we need to know what the acceleration of the object is. We can apply the following equation: vf = vi + a * t, where vf is the final velocity, vi is the initial velocity, a is the acceleration, and t is the time. Since we know all the values except a, we can solve for it by substituting the values. 30 = 0 + a * 120. After simplifying, we get 30 = a * 120. Dividing both sides by 120, we get a = 0.25 m/s^2.

Unveiling the Formula: The Key to the Calculation

Alright, it's time to get our hands dirty with some formulas! The core formula we'll use is derived from the basic principles of motion. Since we're dealing with constant acceleration, we can rely on a neat little equation that directly links distance, initial velocity, time, and acceleration: d = vi*t + 0.5 * a * t^2. The beauty of this equation is that it considers the effect of acceleration over time. The distance the object covers will depend on how fast it starts (initial velocity), how much it speeds up (acceleration), and for how long it speeds up (time).

The equation is our best friend in this case. Now that we know that a = 0.25 m/s^2, t = 120, and vi = 0 m/s, we can substitute it into the formula. Remember that d is displacement (the distance), vi is initial velocity, t is time, and a is acceleration. It is important to remember the units here. Distance is measured in meters, time in seconds, and acceleration in meters per second squared. After substituting the values, we get d = 0*120 + 0.5 * 0.25 * 120^2. Simplifying, we get d = 0 + 0.5 * 0.25 * 14400. d = 1800 meters. And there you have it, folks!

Putting it All Together: Calculating the Final Distance

Time to plug in the numbers and see the magic happen! We now have all the necessary components to calculate the distance. Our known values are: the final velocity (30 m/s), time (120 seconds), and we will assume the initial velocity is zero. To recap, here are the known values:

• Final Velocity (vf): 30 m/s
• Time (t): 120 seconds
• Initial Velocity (vi): 0 m/s

With those values, we know that the acceleration is 0.25 m/s^2. Now we can substitute all the values into the formula to solve for d (displacement). That would be d = vi*t + 0.5 * a * t^2. Remember, the units have to be consistent. Let's substitute the values into the formula to calculate the distance. d = 0*120 + 0.5 * 0.25 * 120^2. First, we can solve what is inside the parentheses, which is 120^2 = 14400. And we can solve 0.5 * 0.25 = 0.125. Which results in the equation: d = 0 + 0.125 * 14400. We know that 0 * 120 = 0. Therefore, the distance (displacement) is d = 1800 meters. This is a very important concept in physics. The final result represents the total distance the object has traveled during the acceleration period. Note that if the initial velocity was not 0 m/s, the total displacement would be different. This makes it important to include all the variables.

So, after a 120-second journey, the mobile object travels a distance of 1800 meters.

Conclusion: The Mobile Object's Triumph

And there you have it, amigos! We've successfully calculated the distance traveled by our mobile object. We started with some basic information, applied the right formulas, and arrived at a solution. This process shows how understanding the principles of physics can help us analyze and predict motion. Remember, every time an object moves, it follows the laws of physics.

This simple problem forms the bedrock for understanding more complex scenarios involving motion. From designing the next generation of rockets to simply predicting how long it will take you to get to the store, these equations are indispensable. So, keep practicing, keep learning, and keep exploring the fascinating world of physics. Until next time, keep those calculations sharp, and your curiosity even sharper! Adios, and happy calculating!