Motion Calculation: Acceleration, Velocity, And Distance

by Sebastian Müller 57 views

Hey guys! Ever wondered how we calculate the movement of an object, like a car speeding up or a ball rolling down a hill? It's all about understanding acceleration, velocity, and distance. Today, we're diving into a fun physics problem that explores these concepts. Let's break it down step by step!

The Problem: A Mobile in Motion

Imagine a mobile, initially at rest, starts moving. After 1.5 minutes, it hits a speed of 35.8 kilometers per hour. Our mission is to figure out three things:

  • a) The acceleration the mobile experienced.
  • b) The velocity of the mobile after 0.8 minutes of travel.
  • c) The distance the mobile covered to reach 35.8 km/h.

Sounds exciting, right? Let's put on our physics hats and get started!

a) Calculating the Acceleration

Understanding Acceleration: In physics, acceleration is the rate at which an object's velocity changes over time. It's not just about going fast; it's about how quickly you're speeding up or slowing down. A car accelerating from a stop sign, a plane taking off, or even a ball rolling down a slope—all these scenarios involve acceleration.

In our problem, the mobile starts from rest, meaning its initial velocity is zero. It then reaches a velocity of 35.8 km/h in 1.5 minutes. To calculate the acceleration, we need to use the following formula:

Acceleration (a) = (Final Velocity (v) - Initial Velocity (u)) / Time (t)

But before we plug in the numbers, there's a crucial step: unit conversion. Physics loves consistency, so we need to ensure all our units are in the same system. The standard unit for velocity in physics calculations is meters per second (m/s), and the standard unit for time is seconds (s). Our final velocity is in kilometers per hour (km/h), and our time is in minutes. Let's convert them!

Converting km/h to m/s: To convert 35.8 km/h to m/s, we multiply by 1000 (to convert kilometers to meters) and divide by 3600 (to convert hours to seconds). This gives us:

35.8 km/h * (1000 m/km) / (3600 s/h) = 9.94 m/s (approximately)

Converting Minutes to Seconds: Converting 1.5 minutes to seconds is straightforward: we multiply by 60:

1.  5 minutes * 60 seconds/minute = 90 seconds

Now we have all our values in the correct units. Let's plug them into our acceleration formula:

a = (9.94 m/s - 0 m/s) / 90 s
a = 0.11 m/s² (approximately)

So, the acceleration the mobile experienced is approximately 0.11 meters per second squared. This means that for every second the mobile moved, its velocity increased by 0.11 m/s. This constant increase in velocity is what we define as uniform or constant acceleration, a crucial concept in understanding motion in physics.

b) Calculating the Velocity after 0.8 Minutes

Velocity at a Specific Time: Now that we know the acceleration, we can figure out the mobile's velocity at any point during its acceleration phase. The question asks for the velocity after 0.8 minutes. We'll use the same acceleration formula, but this time, we're solving for the final velocity.

First, let's convert 0.8 minutes to seconds:

0.8 minutes * 60 seconds/minute = 48 seconds

Now, we can rearrange our acceleration formula to solve for the final velocity (v):

v = u + at

Where:

  • v = Final velocity (what we want to find)
  • u = Initial velocity (0 m/s, since the mobile started at rest)
  • a = Acceleration (0.11 m/s²)
  • t = Time (48 seconds)

Plugging in the values:

v = 0 m/s + (0.11 m/s²) * 48 s
v = 5.28 m/s

Therefore, the velocity of the mobile after 0.8 minutes of travel is 5.28 meters per second. This calculation highlights the relationship between acceleration, time, and velocity. With a constant acceleration, the velocity increases linearly with time. Understanding this relationship is crucial for predicting the motion of objects under constant acceleration.

c) Calculating the Distance Traveled

Distance Under Constant Acceleration: Finally, let's calculate the distance the mobile covered while accelerating to 35.8 km/h. There are a couple of ways to do this, but we'll use the following equation, which is particularly useful when we know the initial velocity, final velocity, acceleration, and we want to find the distance:

v² = u² + 2as

Where:

  • v = Final velocity (9.94 m/s)
  • u = Initial velocity (0 m/s)
  • a = Acceleration (0.11 m/s²)
  • s = Distance (what we want to find)

Rearranging the formula to solve for s:

s = (v² - u²) / (2a)

Plugging in the values:

s = (9.94² - 0²) / (2 * 0.11)
s = 98.8036 / 0.22
s = 449.11 m (approximately)

So, the mobile covered a distance of approximately 449.11 meters while accelerating to 35.8 km/h. This final calculation brings together all the concepts we've discussed: acceleration, velocity, and distance. It demonstrates how these quantities are interconnected and how we can use equations of motion to describe and predict the movement of objects. The distance traveled depends not only on the final speed but also on the acceleration. A higher acceleration means the object reaches the same speed in a shorter time and over a shorter distance.

Wrapping Up

Wow, we did it! We successfully calculated the acceleration, velocity at a specific time, and the distance traveled by our mobile. This problem showcases the fundamental principles of kinematics, the branch of physics that deals with motion. By understanding these principles, we can analyze and predict the movement of objects in various scenarios, from simple everyday occurrences to complex engineering applications.

Remember, physics is all about understanding the world around us. By breaking down problems into smaller steps and applying the right formulas, we can unlock the secrets of motion and much more! Keep exploring, keep questioning, and keep learning, guys!