Problem 63 - Master Resource Book in JEE Main Physics

Question

In an experiment to measure the height of a bridge by dropping stone into water underneath, if the error in measurement of time is \(0.1 \mathrm{~s}\) at the end of \(2 \mathrm{~s}\), then the error in estimation of height of bridge will be [Kerala CEE 2004] (a) \(0.49 \mathrm{~m}\) (b) \(0.98 \mathrm{~m}\) (c) \(1.96 \mathrm{~m}\) (d) \(2.12 \mathrm{~m}\)

Step-by-Step Solution

Answer

The error in estimation of the height is \(1.96 \mathrm{~m}\).

Step 1: Formula for Height Calculation

To find the height of the bridge, we can use the formula for the distance traveled by an object in free fall without initial velocity. The formula is \( h = \frac{1}{2}gt^2 \), where \( g = 9.8 \, \text{m/s}^2 \) and \( t \) is the time in seconds.

Step 2: Calculate Height at Time 2s

Using the formula \( h = \frac{1}{2}gt^2 \), substitute \( g = 9.8 \, \text{m/s}^2 \) and \( t = 2 \, \text{s} \). This gives:\[ h = \frac{1}{2} \times 9.8 \times (2)^2 = \frac{1}{2} \times 9.8 \times 4 = 19.6 \, \text{m} \].

Step 3: Calculate the Rate of Change in Height with Time

The change in height with respect to time is given by the derivative \( \frac{dh}{dt} = gt \), since \( h = \frac{1}{2}gt^2 \). Thus, at \( t = 2 \, \text{s} \), we have:\[ \frac{dh}{dt} = 9.8 \times 2 = 19.6 \].

Step 4: Determine Error in Height Prediction

Given that the error in time \( \Delta t = 0.1 \, \text{s} \), the error in height can be estimated using:\( \Delta h = \frac{dh}{dt} \times \Delta t \). Substitute \( \frac{dh}{dt} = 19.6 \) and \( \Delta t = 0.1 \), we find:\[ \Delta h = 19.6 \times 0.1 = 1.96 \, \text{m} \].

Key Concepts

Understanding Error Analysis

Error analysis is an essential aspect of experiments and calculations in physics. It helps us estimate the possible deviation or error in measurements due to uncertainties. In the example provided, the main focus is on understanding how an error in measuring time can affect the height calculation when an object is in free fall. Consider the scenario where a stone is dropped from a bridge. To determine the height of the bridge, we measure the time it takes for the stone to hit the water. If there is a small error, say a discrepancy of 0.1 seconds in timing, it can significantly impact the calculation of the bridge's height using the kinematic equation for free fall. Key steps in error analysis:

• Identify the source of error: In this case, it's the time measurement.
• Quantify the error: The given error is 0.1 seconds.
• Use the derivative of the function to find the rate at which height changes concerning time: This helps us determine how small mistakes propagate through calculations.
• Estimate the resultant error in the measured value: Calculated using the rate of change of height with respect to time and the error in time measurement.

Derivatives in Physics and their Role

In physics, derivatives help us understand how one quantity changes with respect to another. The use of derivatives is crucial in kinematics and other branches of physics. They describe the object's motion by providing rates of change, such as velocity and acceleration.In the context of our exercise, the derivative \( \frac{dh}{dt} = gt \) indicates how the height changes over time for an object in free fall. It gives us the instantaneous rate of change of height, or how fast the height is changing at any given moment.For example, when \( t = 2 \) seconds, \( \frac{dh}{dt} \) becomes \( 19.6 \) m/s. This means that after 2 seconds of free fall, the height changes at a rate of 19.6 meters for each additional second.By taking the derivative, we find how sensitive our height calculation is to errors in time measurement. The derivative is used to estimate the error in height (\( \Delta h \)) due to error in time (\( \Delta t \)) by multiplying the rate of change with the time error \( \Delta t \). This estimation is a fundamental part of performing error analysis.

Kinematics and Free Fall

Kinematics is a branch of physics that describes the motion of objects without considering the causes of motion. Free fall is a particular case of kinematics where only gravitational force acts on an object, as seen in our example.An object is in free fall when it is only under the influence of gravity, and no other forces (like air resistance) are acting upon it. The formula used in this exercise for determining the height an object has fallen is \( h = \frac{1}{2}gt^2 \). This equation considers:

• \( g \): The acceleration due to gravity (approximately 9.8 m/s² on Earth).
• \( t \): The time duration for which the object has been in free fall.

This formula assumes that the object starts from rest and initial velocity is zero. Therefore, it only accounts for the distance travelled under constant acceleration due to gravity.By substituting values like \( g = 9.8 \, \text{m/s}^2 \) and \( t = 2 \, \text{s} \) into our formula, we find that the stone travels 19.6 meters in 2 seconds. Understanding kinematic equations is key to solving motion problems and forms the basis for more complex physics concepts.