A Ranger In A National Park Is Driving At 46 Km/H When A Dee
A ranger in a national park is driving at 46 km/h when a deer jumps onto the road 86 m ahead of the vehicle. After a reaction time of t s, the ranger applies the brakes to produce an acceleration of −3.8 m/s2. What is the maximum reaction time allowed if the ranger is to avoid hitting the deer? Answer in units of s.
The assignment involves solving physics problems related to motion, including concepts of kinematics such as velocity, acceleration, reaction time, and distance. The specific question asks for the maximum reaction time a ranger can have before hitting a deer on the road, given the initial speed, the distance to the obstacle, and the deceleration applied during braking.
Paper For Above instruction
Introduction
Physics principles underpin many real-world scenarios, particularly in understanding motion and safety measures in transportation. The problem of calculating the maximum reaction time to avoid a collision is a classic application of kinematic equations. It integrates initial velocity, reaction delay, and the deceleration phase to determine safety margins for drivers or operators of vehicles. This analysis emphasizes the importance of reaction time in preventing accidents and highlights the application of fundamental physics formulas in everyday safety scenarios.
Analysis of the Motion
The problem describes a ranger traveling at a constant initial speed of 46 km/h, which needs to be converted into meters per second for compatibility with other units:
\[ v_i = 46\, \text{km/h} = \frac{46 \times 1000}{3600} \approx 12.78\, \text{m/s} \]
The deer is located 86 meters ahead at the moment the ranger spots it. The driver’s reaction time (t) is unknown but must be calculated to ensure the vehicle does not collide with the deer after braking begins. The vehicle then decelerates at a rate of \( a = -3.8\, \text{m/s}^2 \).
The total stopping distance is the sum of the distance traveled during the reaction time and the distance traveled during deceleration.
1. Distance covered during reaction time:
\[ d_{reaction} = v_i \times t \]
2. Distance covered during braking:
Using the equation:
\[ v_f^2 = v_i^2 + 2a d \]
where \( v_f \) is the final velocity (zero at stop), simplifies for deceleration:
\[ 0 = v_i^2 + 2a d_{braking} \]
\[ d_{braking} = -\frac{v_i^2}{2a} \]
Plugging in numbers:
\[ d_{braking} = -\frac{(12.78)^2}{2 \times (-3.8)} \approx \frac{163.23}{7.6} \approx 21.48\, \text{m} \]
The total distance the vehicle covers from reaction to full stop is:
\[ d_{total} = d_{reaction} + d_{braking} \]
\[ d_{total} = v_i t + 21.48\, \text{m} \]
To avoid hitting the deer:
\[ d_{total} \leq 86\, \text{m} \]
\[ 12.78 t + 21.48 \leq 86 \]
\[ 12.78 t \leq 86 - 21.48 \]
\[ 12.78 t \leq 64.52 \]
\[ t \leq \frac{64.52}{12.78} \approx 5.05\, \text{s} \]
Therefore, the maximum reaction time is approximately 5.05 seconds.
Conclusion
This exercise demonstrates the importance of reaction time in vehicle safety and collision avoidance. It quantifies how even a small delay can compromise safety and emphasizes the value of quick reflexes and alertness while driving. The calculated maximum reaction time of approximately 5.05 seconds represents a critical threshold; exceeding it would likely result in a collision with the deer.
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