Problem 17 - Master Resource Book in JEE Main Physics

Question

If the acceleration due to gravity is \(10 \mathrm{~ms}^{-2}\) and the units of length and time are changed in kilometre and hour respectively, the numerical value of acceleration is (a) 360000 (b) 72000 (c) 36000 (d) 129600

Step-by-Step Solution

Answer

The acceleration in kilometer and hour units is 129600.

Step 1: Understanding the Problem

We need to convert the acceleration given in new units. The original acceleration is \( 10 \ \text{m/s}^2 \) with units of meters and seconds. We are required to find its equivalent value when using units of kilometers and hours.

Step 2: Converting Meters to Kilometers

Since 1 kilometer is equivalent to 1000 meters, we first convert the acceleration from meters to kilometers. Thus, \( 1 \text{ meter} = 0.001 \text{ kilometers} \). Hence, 10 \( \text{m/s}^2 = 10 \times 0.001 = 0.01 \text{ km/s}^2 \).

Step 3: Converting Seconds to Hours for Time

There are 3600 seconds in 1 hour (since \( 1 \text{hour} = 60 \text{ minutes} = 3600 \text{ seconds} \)). To convert from \( \text{seconds}^2 \) to \( \text{hours}^2 \), we need this step: \( \text{seconds}^2 \rightarrow \text{hours}^2 = (3600 \text{ s/hour})^2 = 12960000 \text{ s}^2/\text{hour}^2 \).

Step 4: Calculating the Conversion

Now, calculate the equivalent acceleration in \( \text{km/h}^2 \):\[\text{Acceleration in km/h}^2 = 0.01 \text{ km/s}^2 \times 12960000 \text{ s}^2/\text{hour}^2 = 129600 \text{ km/h}^2.\]

Step 5: Choice Selection

The calculated value is 129600, which corresponds to option (d).

Key Concepts

Acceleration Due to Gravity

When we talk about acceleration due to gravity, we're discussing the rate at which an object speeds up as it falls freely toward the earth. This acceleration is constant at any given location on the earth's surface and is generally rounded to 9.8 meters per second squared ( \(9.8 \, \text{m/s}^2\) ).
For simplified problems, you might see it approximated to \(10 \, \text{m/s}^2\).
This means that every second, an object in free fall will gain \(10 \, \text{m/s}\) in velocity.

• It's a constant that results from the gravitational force exerted by the Earth.
• Used in all calculations involving objects in free fall.
• Effective at any point close to the Earth's surface.

Understanding gravity helps us predict how fast an object will move when dropped. This fundamental concept is crucial in physics as it applies to a myriad of problems and real-world scenarios.

SI Units

SI, or the International System of Units, is the standard metric system used worldwide. It consists of seven base units, such as the meter for length and the second for time.
In the context of physics, using standard SI units ensures consistency and accuracy across all scientific fields.

• Length: Meter (m) - the base unit for measuring distance.
• Time: Second (s) - the base unit for measuring time.
• Mass: Kilogram (kg) - the base unit for measuring mass.

SI units are essential because they provide a clear and coherent way to express measurements, which is crucial for performing precise calculations and comparing results effectively. They are universally accepted, eliminating confusion and enhancing communication in science and engineering.

Conversion of Units

Conversion of units is a crucial skill in physics and science in general. It involves changing a measurement from one unit to another to maintain consistency in calculations. For instance, converting the acceleration value from \(\text{m/s}^2\) to \(\text{km/h}^2\) requires understanding and applying appropriate factors for both length and time.

• Length Conversion: Recognize the ratio, for example, 1 kilometer equals 1000 meters.
• Time Conversion: Know that 1 hour equals 3600 seconds.
• Formula Application: Combine unit conversions to adapt more complex measures like acceleration.

The process is straightforward: multiply by activities like 1000 to convert meters to kilometers and by 3600 squared to convert \(\text{seconds}^2\) to \(\text{hours}^2\). This ensures that all physical quantities are expressed in terms that are most convenient for specific calculations.