Suppose that a woman driving a Mercedes zooms out of a darkened tunnel at 33.9 m/s. She is momentarily blinded by the sunshine. When she recovers, she sees that she is fast overtaking a bus ahead in her lane moving at the slower speed of 12.0 m/s. She hits the brakes as fast as she can (her reaction time is 0.35 s). If she can decelerate at 2.9 m/s2, what is the minimum distance between the driver and the bus when she first sees it so that they do not collide?
Well, it sounds like the woman in the Mercedes is in quite the predicament! Let's see if we can figure out the distance between her and the bus to avoid a collision.
First, let's calculate how far the woman travels during her reaction time of 0.35 s. Since she is moving at a speed of 33.9 m/s, we can use the equation distance = speed × time to find out. Therefore, the distance she travels during her reaction time is 33.9 m/s × 0.35 s = 11.865 m.
Now, let's figure out her deceleration distance. We can use the equation final velocity squared = initial velocity squared + 2 × acceleration × distance. Her final velocity is 12.0 m/s (the same as the bus), her initial velocity is 33.9 m/s, and her acceleration is -2.9 m/s^2 (negative because she's decelerating). Rearranging the equation, we get distance = (final velocity squared - initial velocity squared) / (2 × acceleration). Plugging in the values, we have distance = (12.0 m/s)^2 - (33.9 m/s)^2 / (2 × -2.9 m/s^2) ≈ 2.44 m.
Finally, let's add the distance traveled during her reaction time to the deceleration distance to find the minimum distance between the driver and the bus. That is 11.865 m + 2.44 m ≈ 14.305 m.
So, the minimum distance between the driver and the bus for them to avoid a collision is approximately 14.305 meters. Stay safe on the roads and try not to blind your fellow drivers with your fancy cars and their shiny grilles!
To find the minimum distance between the driver and the bus when she first sees it so that they do not collide, we need to calculate the stopping distance for the Mercedes and determine the distance traveled during the driver's reaction time.
Step 1: Calculate the distance traveled during the driver's reaction time.
Given:
Initial speed (vi) = 33.9 m/s
Reaction time (t) = 0.35 s
During the reaction time, the Mercedes will cover a distance equal to the distance traveled at the initial speed (vi) during the reaction time (t):
Distance during reaction time = vi * t
Substituting the values:
Distance during reaction time = 33.9 m/s * 0.35 s = 11.865 m
Step 2: Calculate the stopping distance for the Mercedes.
Given:
Deceleration (a) = -2.9 m/s^2 (negative as it represents deceleration)
The stopping distance can be calculated using the following equation:
Stopping distance = (vf^2 - vi^2) / (2 * a)
where:
vf = final speed (which would be 0 m/s as the Mercedes comes to a stop)
Substituting the values:
Stopping distance = (0 - (33.9^2)) / (2 * -2.9)
Stopping distance = (0 - 1147.21) / -5.8
Stopping distance = 1147.21 / 5.8
Stopping distance = 197.78 m (rounded to two decimal places)
Step 3: Calculate the minimum distance between the driver and the bus.
Given:
Speed of the bus (vb) = 12.0 m/s
The minimum distance between the driver and the bus will be the sum of the distance traveled during the reaction time and the stopping distance for the Mercedes:
Minimum distance = Distance during reaction time + Stopping distance
Minimum distance = 11.865 m + 197.78 m
Minimum distance = 209.645 m (rounded to three decimal places)
Therefore, the minimum distance between the driver and the bus when she first sees it so that they do not collide is approximately 209.645 meters.
To find the minimum distance between the driver and the bus, we need to determine the distance the driver travels during her reaction time and the distance it takes for her to come to a stop.
Let's break down the problem step by step:
Step 1: Find the distance traveled during the driver's reaction time.
During the reaction time, the driver maintains her initial speed of 33.9 m/s. To calculate the distance traveled during the reaction time, we use the formula:
Distance = Speed x Time
The time is given as 0.35 s, and the speed is 33.9 m/s. Therefore, the distance traveled during the reaction time is:
Distance = 33.9 m/s x 0.35 s = 11.865 m.
Step 2: Calculate the distance traveled while decelerating to a stop.
To calculate the distance traveled while decelerating, we use the equation:
Distance = (Initial Velocity^2 - Final Velocity^2) / (2 x Acceleration)
The initial velocity is 33.9 m/s, the final velocity is 0 m/s (as the car comes to a stop), and the acceleration is -2.9 m/s^2 (negative as the car is decelerating).
Plugging in the values:
Distance = (33.9 m/s)^2 - (0 m/s)^2 / (2 x -2.9 m/s^2)
Distance = 33.9 m/s^2 / (2 x -2.9 m/s^2)
Distance = 33.9 m/s^2 / -5.8 m/s^2
Distance = -5.8448 m.
Note: The negative sign indicates that the car is moving in the opposite direction (opposite direction to its velocity).
Step 3: Find the total minimum distance.
To find the total minimum distance between the driver and the bus, we add the distances from Step 1 and Step 2:
Total Minimum Distance = Distance traveled during reaction time + Distance traveled while decelerating
Total Minimum Distance = 11.865 m + (-5.8448 m)
Total Minimum Distance = 6.0202 m
Therefore, the minimum distance between the driver and the bus when she first sees it so that they do not collide is approximately 6.02 meters.