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.What is her speed as she reaches the light at the instant it turns green?
Well, it sounds like this motorist has quite the timing challenge on their hands! Let's see if we can figure out how fast they need to be going.
First, let's calculate how much distance the motorist will cover during the 15-second red light period. Since she's initially going at 20 m/s and the light stays red for 15 seconds, she will cover a distance of 20 m/s * 15 s = 300 meters during that time.
Now, let's figure out the deceleration. The motorist takes 1.0 second to step on the brakes and begin slowing down. Let's say her final velocity when she reaches the light is V, and her initial velocity is 20 m/s. We can use the formula:
V = U + at
where V is the final velocity, U is the initial velocity, a is the deceleration, and t is the time.
Since U = 20 m/s, t = 1.0 s, and we want to find out V, we can rearrange the formula to solve for a:
a = (V - U) / t
Now, remember that the motorist wants to reach the light just as it turns green again. This means they need to cover a total distance of 300 m, which is the distance during the red light period.
We can use the kinematic equation:
s = ut + 0.5at^2
where s is the distance, u is the initial velocity, a is the deceleration, and t is the time.
Rearranging this equation to solve for a, we get:
a = 2(s - ut) / t^2
Using s = 300 m, u = 20 m/s, and t = 1.0 s, we can calculate a.
Now that we know the deceleration, we can use the formula:
V^2 = U^2 + 2as
where V is the final velocity, U is the initial velocity, a is the deceleration, and s is the distance.
Again, U = 20 m/s, a is the deceleration we just calculated, and s = 300 m.
Plug in these values and solve for V to find out the motorist's speed as she reaches the light at the instant it turns green.
To find the motorist's speed as she reaches the light at the instant it turns green, we need to calculate the distance traveled during the deceleration and the time it takes to reach the light.
1. Calculate the distance traveled during the deceleration:
During the deceleration, the motorist is slowing down from 20 m/s to 0 m/s. The deceleration is assumed to be constant, and it will take 1.0 second for the motorist to stop.
Using the following equation of motion:
v = u + at
Where:
u = 20 m/s (initial velocity)
v = 0 m/s (final velocity)
t = 1.0 s (time taken)
Rearranging the equation:
a = (v - u) / t
a = (0 - 20) / 1.0
a = -20 m/s^2
Now, we can use the equation of motion to calculate the distance traveled during deceleration:
s = ut + (1/2)at^2
Where:
s = distance
u = initial velocity
t = time taken
a = acceleration
s = (20)(1.0) + (1/2)(-20)(1.0)^2
s = 20 - 10
s = 10 meters
Therefore, the distance traveled during the deceleration is 10 meters.
2. Calculate the time taken to reach the light:
The motorist wants to reach the light just as it turns green again. The traffic light stays red for 15 seconds. Therefore, the motorist needs to reach the light 15 seconds after it turned red.
The total time taken by the motorist to reach the light is the sum of the deceleration time (1.0 second) and the time the light stays red (15 seconds).
Total time taken = 1.0 + 15
Total time taken = 16 seconds
3. Calculate the speed as the motorist reaches the light:
To calculate the speed, we need to find the average velocity during the total time taken.
Average velocity = total distance traveled / total time taken
The total distance traveled is the sum of the distance during deceleration (10 meters) and the distance traveled at a constant velocity during the time the light stays red.
The constant velocity is the same as the initial velocity of the motorist, which is 20 m/s.
Total distance traveled = distance during deceleration + (velocity)(time) during the red light
Total distance traveled = 10 + (20 m/s)(15 s)
Total distance traveled = 10 + 300
Total distance traveled = 310 meters
Average velocity = 310 meters / 16 seconds
Average velocity = 19.375 m/s
Therefore, the motorist's speed as she reaches the light at the instant it turns green is approximately 19.375 m/s.
To find the motorist's speed as she reaches the light at the instant it turns green, we can follow these steps:
Step 1: Calculate the time it takes for the motorist to reach the light after seeing it turn red.
Distance = Speed × Time
Given:
Distance = 200 m
Speed = 20 m/s
Time = ?
Rearranging the formula: Time = Distance / Speed
Time = 200 m / 20 m/s
Time = 10 s
Step 2: Subtract the red light duration from the time to reach the light.
Total Time = Time to reach the light - Red light duration
Total Time = 10 s - 15 s
Total Time = -5 s
Step 3: Calculate the time it takes for the motorist to stop after stepping on the brakes.
Given:
Initial Speed = 20 m/s
Final Speed = 0 m/s
Time = 1.0 s
Rearrange the formula: Acceleration = (Final Speed - Initial Speed) / Time
Acceleration = (0 m/s - 20 m/s) / 1.0 s
Acceleration = -20 m/s^2
Step 4: Calculate the distance traveled while decelerating.
Using the formula for uniformly decelerated motion: Distance = Initial Speed × Time + 0.5 × Acceleration × Time^2
Distance = 20 m/s × 1.0 s + 0.5 × (-20 m/s^2) × (1.0 s)^2
Distance = 20 m/s - 10 m/s^2 × 1.0 s^2
Distance = 20 m - 10 m
Distance = 10 m
Step 5: Calculate the motorist's speed as she reaches the light at the instant it turns green.
Remaining Distance = Total Distance (200 m) - Distance traveled while decelerating (10 m)
Remaining Distance = 200 m - 10 m
Remaining Distance = 190 m
Speed = Remaining Distance / Total Time
Speed = 190 m / (-5 s)
Speed = -38 m/s
The motorist's speed as she reaches the light at the instant it turns green is -38 m/s. Note that the negative sign indicates that the motorist is moving in the opposite direction, or that she has overshot the light.