A driver of a car going 95.0 km/h suddenly sees the lights of a barrier 39.0 m ahead. It takes the driver 0.95 s before he applies the brakes, and the average acceleration during braking is -10.0 m/s2.
What is the maximum speed at which the car could be moving and not hit the barrier 39.0 m ahead? Assume the acceleration rate doesn't change.
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To find the maximum speed at which the car could be moving and not hit the barrier, we need to calculate the distance the car travels during the reaction time and the distance it can travel while decelerating.
1. Calculate the distance traveled during the reaction time:
The reaction time is given as 0.95 s. Since the car is initially moving at a constant speed, we can use the formula: distance = speed × time.
So, the distance traveled during the reaction time is: distance = 95.0 km/h × (0.95 s) × (1000 m/1 km) × (1 h/3600 s).
Converting units, we have: distance = (95.0 × 1000 × 0.95) m/3600.
Simplifying, we get: distance = 25.694 m.
2. Calculate the remaining distance the car can travel while decelerating:
To calculate the remaining distance, we use the equation: distance = initial velocity × time + (1/2) × acceleration × time^2.
Here, the initial velocity is the speed at which the car was moving before applying the brakes, and the time is the reaction time (0.95 s).
The final velocity during braking is 0 m/s as the car comes to a stop.
Rearranging the equation, we get: distance = (initial velocity × time) + (1/2) × acceleration × time^2.
Since we want to find the remaining distance, we can rearrange the equation again: distance = (1/2) × acceleration × time^2.
Substituting the given values, we have: distance = (1/2) × (-10.0 m/s^2) × (0.95 s)^2.
Simplifying, we get: distance = (1/2) × (-10.0) × (0.95^2).
Calculating, we get: distance ≈ -4.28 m.
3. Calculate the maximum speed at which the car could be moving without hitting the barrier:
To find the maximum speed, we subtract the distances calculated in steps 1 and 2 from the total distance to the barrier.
Total distance = distance traveled during reaction time + remaining distance.
Total distance = 39.0 m - 25.694 m - (-4.28 m).
Simplifying, we get: Total distance = 39.0 m - 25.694 m + 4.28 m.
Calculating, we get: Total distance ≈ 17.586 m.
Now, we have the total distance the car can travel before it hits the barrier. To find the maximum speed, we use the formula: speed = distance / time.
Here, the distance is 17.586 m (the remaining distance) and the time is the reaction time (0.95 s).
So, the maximum speed at which the car could be moving without hitting the barrier is: speed = 17.586 m / 0.95 s.
Calculating, we get: speed ≈ 18.519 m/s.
Therefore, the maximum speed at which the car could be moving without hitting the barrier is approximately 18.519 m/s.