A motorist is driving at 20m/s{\rm m/s} when she sees that a traffic light 200m{\rm m} ahead has just turned red. She knows that this light stays red for 15s{\rm s} , and she wants to reach the light just as it turns green again. It takes her 1.0s{\rm s} to step on the brakes and begin slowing at a constant deceleration.
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To calculate the required deceleration for the motorist to reach the traffic light just as it turns green again, we can use the following steps:
Step 1: Calculate the distance covered by the motorist before she starts to decelerate.
The initial velocity of the motorist is 20 m/s, and she takes 1.0 s to step on the brakes. Therefore, the distance covered during this time can be calculated using the formula:
Distance = Initial Velocity * Time
Distance = 20 m/s * 1.0 s
Distance = 20 m
Step 2: Calculate the remaining distance to cover before reaching the traffic light.
The total distance to the traffic light is given as 200 m. Therefore, the remaining distance after the initial distance covered can be calculated as:
Remaining Distance = Total Distance - Initial Distance
Remaining Distance = 200 m - 20 m
Remaining Distance = 180 m
Step 3: Calculate the time required to cover the remaining distance with a constant deceleration.
The equation to calculate the distance covered with a constant deceleration is given by:
Distance = (Initial Velocity * Time) + (0.5 * Deceleration * Time^2)
Here, the initial velocity is 0 m/s (since the motorist wants to stop), and the remaining distance is 180 m. We need to determine the deceleration and time required.
0 m = (0 m/s * Time) + (0.5 * Deceleration * Time^2)
Since the motorist wants to reach the traffic light just as it turns green again, the time required would be 15 s. Plugging this value into the equation:
0 m = (0 m/s * 15 s) + (0.5 * Deceleration * (15 s)^2)
Simplifying the equation:
0 = 0 + (0.5 * Deceleration * 225 s^2)
0 = 112.5 * Deceleration * s^2
Therefore, Deceleration = 0, indicating that the motorist needs to apply the brakes with zero deceleration in order to reach the traffic light just as it turns green again.
To find the deceleration required to stop the car at the traffic light just as it turns green again, we can use the kinematic equation:
v^2 = u^2 + 2as
Where:
v = final velocity of the car (0 m/s, since it needs to stop)
u = initial velocity of the car (20 m/s)
a = deceleration (what we need to find)
s = distance traveled by the car (200 m)
Rearranging the equation and substituting in the given values:
0^2 = 20^2 + 2a(200)
0 = 400 + 400a
400a = -400
a = -1 m/s^2
Therefore, the deceleration needed to stop the car just as the traffic light turns green again is -1 m/s^2.