Problem 5 - Master Resource Book in JEE Main Physics

Question

In a race for \(100 \mathrm{~m}\) dash, the first and the second runners have a gap of one metre at the mid way stage. Assuming the first runner goes steady, by what percentage should the second runner increases his speed just to win the race. (a) \(2 \%\) (b) \(4 \%\) (c) more than \(4 \%\) (d) less than 496

Step-by-Step Solution

Answer

(c) more than 4\%

Step 1: Understand the problem

The problem involves two runners in a 100-meter dash. At the 50-meter (midway) point, the first runner is 1 meter ahead of the second runner, meaning the first runner has covered 50 meters and the second runner has covered 49 meters. We need to determine how much the second runner needs to increase his speed to win the race.

Step 2: Determine remaining distance

At the midway point, there are still 50 meters remaining for the first runner, while the second runner still has 51 meters to cover (49 meters covered so far).

Step 3: Calculate the time for the first runner

Assume the first runner completes 50 meters in time \( t \) with a constant speed. Hence, his speed \( v_1 \) is \( \frac{50}{t} \).

Step 4: Determine time available for the second runner

For the second runner to win by a small margin, he must complete his 51 meters in time \( t \), the same time taken by the first runner to complete the remaining 50 meters.

Step 5: Calculate required speed for the second runner

The speed required for the second runner, \( v_2 \), is given by \( \frac{51}{t} \).

Step 6: Determine the original speed of the second runner

The original speed of the second runner was \( \frac{49}{t} \), as he covered 49 meters in the time the first runner covered 50 meters.

Step 7: Calculate required speed increase percentage

To find the percentage increase in speed, use the formula:\[ \frac{(v_2 - \text{original speed})}{\text{original speed}} \times 100 \]\[ \text{Speed increase percentage} = \frac{\left(\frac{51}{t} - \frac{49}{t}\right)}{\frac{49}{t}} \times 100 \]Simplifying gives:\[ \frac{(2/49)}{1} \times 100 \approx 4.08\% \]

Step 8: Finalize the choice

The percentage increase required for the second runner to win is approximately \(4.08\%\), which is more than \(4\%\). Therefore, the correct answer is (c) more than \(4\%\).

Key Concepts

Kinematics

Kinematics is a branch of mechanics that deals with the motion of objects without considering the forces that cause these motions. In our exercise, we are dealing with kinematics through a 100-meter dash race, focusing on concepts such as distance, speed, and time.
Kinematics often involves:

• Describing motion through graphs and equations
• Understanding uniform (constant) motion versus uniformly accelerated motion
• Relating displacement, velocity, and acceleration with time

In a race, when we mention that a runner "goes steady," we imply that the runner maintains a constant speed. Speed, one of the vital kinematic quantities, is calculated here using the simple relation between distance and time: \[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \]Understanding this relation helps us write the time equations we need when determining how quickly each runner moves during the race.

Relative Speed Calculation

Relative speed refers to the speed of one object as observed from another object. In this exercise, it helps us compare the speeds of two runners during the race.
The concept of relative speed becomes essential when understanding the race dynamics, as it indicates how fast one runner is gaining on another. Here, it’s crucial to comprehend that although both runners may have the same baseline speed, the relative positioning (e.g., one runner being 1 meter ahead) alters their required speeds to achieve their next goal.
To calculate relative speed effectively:

• For runners in the same direction, subtract their speeds
• Translate the remaining distance into necessary speed variations

In our scenario, to calculate how the second runner adjusts his speed, we determine the first's runner speed and compare it against the second's, focusing on the final distances to be covered (51 meters more for the second runner versus 50 meters for the first). This comparison helps frame the necessary speed increase for the trailing runner.

Percentage Increase in Speed

When talking about percentage increase in speed, we’re evaluating how much faster one runner needs to be relative to their original speed to achieve a certain target.
In our exercise, we want to know by how much the second runner needs to boost his speed to overtake the first. The calculation approach involves:

• Identifying the original speed of the runner
• Calculating the required speed post-increase, so they finish the race in the same (or lesser) time as the first runner
• Using the formula for percentage increase: \[ \text{Percentage Increase} = \frac{(\text{New Speed} - \text{Original Speed})}{\text{Original Speed}} \times 100 \]

In this specific problem, with original speed being \( \frac{49}{t} \) and new speed needed as \( \frac{51}{t} \), the increase calculates to about 4.08\%, confirming the runner must increase his speed by more than 4\% to win. This understanding of percentage change is crucial, especially in competitive situations, to determine feasibilities and strategic decisions.