In physics, motion is a fundamental concept that describes how objects move over time. One important aspect of motion is the ratio of distances traversed in successive intervals. This concept is crucial in understanding uniform acceleration, kinematic equations, and motion analysis.
When an object moves under constant acceleration, the distances it covers in equal time intervals follow a specific pattern. This principle is widely used in fields like mechanics, engineering, and astronomy. In this topic, we will explore how distances in successive intervals relate to each other, the mathematical derivation, and real-world applications.
Understanding Motion in Successive Intervals
1. Motion Under Uniform Acceleration
When an object moves with constant acceleration, its velocity increases at a steady rate. The distances covered in successive time intervals form a specific ratio. This is particularly relevant in free-fall motion, vehicle acceleration, and projectile motion.
2. Equation of Motion
Using kinematic equations, the total distance covered in n seconds under constant acceleration is given by:
where:
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** S_n ** = Distance covered in n seconds
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** u ** = Initial velocity
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** a ** = Acceleration
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** n ** = Time interval
To find the ratio of distances traversed in successive seconds, we analyze the distances covered in the 1st, 2nd, 3rd, … nth seconds.
Ratio of Distances in Successive Intervals
1. Distances Covered in Successive Seconds
The distance covered in the n-th second is given by the formula:
For an object starting from rest ( u = 0 ), the distances covered in the first few seconds are:
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First second: S_1 = frac{1}{2} a (2 times 1 – 1) = frac{1}{2} a
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Second second: S_2 = frac{1}{2} a (2 times 2 – 1) = frac{3}{2} a
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Third second: S_3 = frac{1}{2} a (2 times 3 – 1) = frac{5}{2} a
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Fourth second: S_4 = frac{1}{2} a (2 times 4 – 1) = frac{7}{2} a
2. Ratio of Successive Distances
The ratio of distances covered in successive seconds is:
This follows the pattern 1, 3, 5, 7, … (an arithmetic sequence with a common difference of 2).
Thus, for an object in uniformly accelerated motion, the distances covered in successive seconds follow the ratio 1:3:5:7:9….
Why Does This Ratio Occur?
1. Relationship Between Acceleration and Distance
Acceleration ( a ) affects the rate at which velocity changes. Since velocity increases linearly with time ( v = u + at ), the distance covered in each second grows quadratically. This results in the odd-numbered ratio seen in successive intervals.
2. Free-Fall Motion Example
In free-fall motion, an object dropped from a height follows this exact pattern. If a ball is dropped from a building, the distances it falls in the first few seconds follow the ratio 1:3:5:7.
Real-World Applications
1. Free-Falling Objects
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When an object is dropped from a height, it follows gravitational acceleration ( g approx 9.8 , text{m/s}^2 ).
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The distances covered in each second follow the 1:3:5:7 pattern.
2. Car Acceleration
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A car accelerating from rest covers increasing distances in each second.
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If acceleration remains constant, the distances in successive seconds will follow the same ratio.
3. Projectile Motion
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When a ball is thrown vertically, its motion during ascent and descent follows this principle.
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Understanding these ratios helps in predicting motion paths.
4. Engineering and Space Science
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Engineers use these concepts to design roller coasters, space launch sequences, and braking systems.
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In space travel, this principle helps in calculating velocity changes for orbital maneuvers.
Comparing Uniform and Non-Uniform Motion
| Type of Motion | Distance Covered in Successive Seconds | Example |
|---|---|---|
| Uniform Motion | Equal distances in each second | A car moving at constant speed |
| Uniform Acceleration | Ratio follows 1:3:5:7… | A falling ball, accelerating car |
| Non-Uniform Acceleration | Unpredictable distances | A rocket launching with changing thrust |
This table highlights the importance of understanding motion types in physics and engineering.
Common Misconceptions
1. “Objects fall at constant speed”
Incorrect! Objects in free fall accelerate due to gravity. The distances covered increase over time.
2. “Velocity and distance increase at the same rate”
Velocity increases linearly, but distance increases quadratically, leading to the 1:3:5:7 pattern.
3. “The ratio applies to all motion types”
This ratio applies only to uniform acceleration. For varying acceleration, the pattern changes.
Frequently Asked Questions (FAQs)
Q1: Why do objects cover more distance in each successive second?
Because velocity increases over time under constant acceleration, leading to greater displacement in each second.
Q2: How does this principle apply to sports?
In sprinting, runners start slow and increase speed, covering more distance per second as they accelerate.
Q3: Does this ratio apply to deceleration?
Yes! When an object slows down uniformly, the distances covered decrease in the reverse ratio (e.g., 7:5:3:1).
Q4: Can we use this principle in vehicle safety?
Yes! Engineers use it to design braking systems, airbags, and crash simulations to predict how cars stop under acceleration changes.
Q5: How does gravity affect this ratio?
Gravity provides a constant acceleration of 9.8 m/s², ensuring that freely falling objects follow the 1:3:5:7 distance ratio.
The ratio of distances traversed in successive intervals plays a fundamental role in understanding motion under uniform acceleration. It explains how objects move in real-world scenarios like free fall, vehicle acceleration, and projectile motion.
Key Takeaways:
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Objects moving under constant acceleration cover distances in a 1:3:5:7 ratio.
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The formula ** S_n = u + frac{1}{2} a (2n – 1) ** describes motion in successive seconds.
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This principle applies to falling objects, accelerating cars, and space travel.
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Engineers and scientists use this concept in transportation, sports, and aerospace design.
By understanding this motion pattern, we can predict and optimize how objects move under acceleration, making this concept valuable in both everyday life and advanced physics applications.