Question

A particle has an acceleration of \(72 \mathrm{~km} / \mathrm{min}^{2}\) find acceleration in SI system. (a) \(0.5 \mathrm{~m} / \mathrm{s}^{2}\) (b) \(30 \mathrm{~m} / \mathrm{s}^{2}\) (c) \(18 \mathrm{~m} / \mathrm{s}^{2}\) (d) \(20 \mathrm{~m} / \mathrm{s}^{2}\)

Short answer

The correct answer is (d) \(20 \mathrm{~m/s}^{2}\).

Step-by-step solution

Identify the given acceleration in km/min²

The given acceleration is 72 km/min².

Convert km to meters

We know that 1 km = 1000 m. To convert the given acceleration from km to meters, we'll multiply by 1000. \(72 \mathrm{~km} \times 1000 = 72000 \mathrm{~m}\) So, the acceleration is now \(72000 \mathrm{~m/min}^{2}\).

Convert minutes to seconds

We know that 1 minute = 60 seconds. To convert acceleration from m/min² to m/s², we'll divide by the square of 60. (As the time is squared) \(72000 \mathrm{~m/min}^{2} \div (60 \times 60) = 72000 \mathrm{~m/min}^{2} \div 3600 = 20 \mathrm{~m/s}^{2}\)

Find the correct option

Now that we have found the acceleration in the SI system, which is \(20 \mathrm{~m/s}^{2}\), we can see that the correct option is: (d) \(20 \mathrm{~m/s}^{2}\)

Key concepts

Acceleration

Acceleration is a measure of how quickly an object changes its velocity. It tells us how fast the speed of something is increasing or decreasing. Imagine you're in a car. If the car suddenly speeds up, that change is due to acceleration. Similarly, if you slam on the brakes, the deceleration is also a form of acceleration, but in the opposite direction.

Key aspects of acceleration include:

• Direction: Acceleration involves a change in speed and direction. It's a vector, meaning it has both magnitude and direction.
• Units: Acceleration is often measured in meters per second squared (m/s²) in the SI system. This indicates how much the velocity changes each second.

Understanding acceleration helps us comprehend motion better, whether it's a particle moving on a straight line or a spaceship soaring through space.

SI Units

The International System of Units (SI) is the standard system for measuring physical quantities globally. Using SI units ensures everyone uses the same measurement system, which simplifies science and engineering.

When talking about acceleration, the SI unit is meters per second squared (m/s²). Here's how we got there from kilometers per minute squared:

• Kilometers to Meters: Since 1 kilometer is 1000 meters, we multiply by 1000 to convert.
• Minutes to Seconds: Since 1 minute is 60 seconds, and time is squared in this case, we divide by 60² or 3600.

This standardization not only aids in calculations but also enhances communication, ensuring that scientific findings are universally understood.

Dimensional Analysis

Dimensional analysis is a powerful method used to convert one unit of measure to another. It allows us to check if an equation makes sense based on its units. By ensuring that both sides of an equation involve the same dimensions, we can verify accuracy without complex calculations.

In our exercise, dimensional analysis helped us convert kilometers per minute squared to meters per second squared. Here's a quick breakdown:

• Length Dimension: Convert from kilometers to meters by multiplying by 1000.
• Time Dimension: As time is squared, convert minutes to seconds by dividing by 3600.

Using dimensional analysis not only guides these conversions but also ensures that the math aligns with physical realities. It's like a sanity check for equations, ensuring they remain logical and applicable.