A driver of a car going 90 km/h suddenly sees the lights of a barrier 40.0 m ahead. It takes the driver 0.75 s to apply the brakes, and the average acceleration during braking is -10 m/s2. (a) Compute for the total displacement of the car to determine whether the car hits the barrier. (b) At what maximum speed should the car be initially moving so as not to hit the barrier 40 m ahead? Assume that the reaction time and acceleration given above does not change.
2.5
To answer these questions, we need to analyze the motion of the car and calculate the displacement in both cases.
Let's start with part (a).
(a) To determine whether the car hits the barrier, we need to calculate the total displacement of the car during the braking process. The total displacement can be calculated by finding the sum of the distances covered during the reaction time and the braking time.
During the reaction time of 0.75 seconds, the car is still moving at a speed of 90 km/h = (90 * 1000) m/3600 s = 25 m/s. Therefore, the distance covered during the reaction time is:
Distance during reaction time = speed * time = 25 m/s * 0.75 s = 18.75 m
The car then starts decelerating with an acceleration of -10 m/s^2. We can calculate the distance covered during the braking time using the equation:
Distance during braking time = (initial speed * time) + (0.5 * acceleration * time^2)
Given that the initial speed is 25 m/s, and the time to brake is 0.75 s, we can calculate the distance during braking as:
Distance during braking time = (25 m/s * 0.75 s) + (0.5 * (-10 m/s^2) * (0.75 s)^2)
= 18.75 m + (-2.8125 m)
= 15.9375 m
The total displacement of the car is the sum of the distances covered during the reaction time and braking time:
Total displacement = Distance during reaction time + Distance during braking time
= 18.75 m + 15.9375 m
= 34.6875 m
Since the total displacement of the car is less than the distance to the barrier (40 m), the car does not hit the barrier.
(b) Now, let's calculate the maximum speed at which the car should be initially moving to avoid hitting the barrier.
In this case, the car must come to a stop just as it reaches the barrier. Therefore, the total displacement of the car during the braking process should be equal to the distance to the barrier (40 m).
From part (a), we found that the total displacement during the braking process is 34.6875 m. So, we need to find the initial speed that will result in a total displacement of 40 m.
Using the same equation as in part (a), we can rearrange it to solve for the initial speed:
Distance during braking time = (initial speed * time) + (0.5 * acceleration * time^2)
Substituting the known values, we get:
40 m = (initial speed * 0.75 s) + (0.5 * (-10 m/s^2) * (0.75 s)^2)
Simplifying the equation, we have:
40 m = (initial speed * 0.75 s) - 2.8125 m
(initial speed * 0.75 s) = 42.8125 m
initial speed = 42.8125 m / 0.75 s
initial speed ≈ 57.1 m/s
Therefore, to avoid hitting the barrier, the car should be initially moving at a maximum speed of approximately 57.1 m/s or 205.56 km/h.
(a) To compute the total displacement of the car and determine whether the car hits the barrier, we need to calculate the distance the car travels during the reaction time and during the braking period.
1. Distance during reaction time:
The formula to calculate distance is: distance = initial velocity * time
The initial velocity of the car is 90 km/h, which needs to be converted to m/s:
90 km/h = 90 * 1000 m/3600 s = 25 m/s
During the reaction time of 0.75 s, the car will travel a distance of:
distance = 25 m/s * 0.75 s = 18.75 m
2. Distance during braking:
The formula to calculate distance is: distance = initial velocity * time + (1/2) * acceleration * time^2
The initial velocity during the braking period is 25 m/s (from the reaction time calculation).
The acceleration is -10 m/s^2.
We need to find the time it takes for the car to come to a stop. To do this, we can use the formula:
final velocity = initial velocity + acceleration * time
0 m/s = 25 m/s + (-10 m/s^2) * time
Solving for time:
10 m/s^2 * time = 25 m/s
time = 25 m/s / 10 m/s^2
time = 2.5 s
Now, we can calculate the distance during braking:
distance = initial velocity * time + (1/2) * acceleration * time^2
distance = 25 m/s * 2.5 s + (1/2) * (-10 m/s^2) * (2.5 s)^2
distance = 62.5 m - 31.25 m
distance = 31.25 m
3. Total displacement:
The total displacement is the sum of the distances during the reaction time and the braking period:
total displacement = distance during reaction time + distance during braking
total displacement = 18.75 m + 31.25 m
total displacement = 50 m
The total displacement of the car is 50 meters. Since this is greater than the distance to the barrier (40 meters), the car will hit the barrier.
(b) To find the maximum initial speed at which the car will not hit the barrier, we can use the same steps as above, but this time we need to find the maximum initial velocity.
We know that the total displacement needs to be equal to the distance to the barrier (40 meters).
Setting up the equation:
total displacement = distance during reaction time + distance during braking
40 m = (25 m/s * 0.75 s) + (25 m/s * time) + (1/2) * (-10 m/s^2) * time^2
Solving for time:
10 m/s * time^2 + 25 m/s * time - 16.25 m = 0
We can solve this quadratic equation using the quadratic formula:
time = (-b ± sqrt(b^2 - 4ac)) / 2a
Plugging in the values:
time = [-(25 m/s) ± sqrt((25 m/s)^2 - 4 * 10 m/s * (-16.25 m))] / (2 * 10 m/s)
Solving for time using the positive root:
time ≈ 1.19 s
Now, we can calculate the maximum initial velocity:
initial velocity = (total displacement - distance during reaction time - (1/2) * acceleration * time^2) / time
initial velocity = (40 m - (25 m/s * 0.75 s) - (1/2) * (-10 m/s^2) * (1.19 s)^2) / 1.19 s
initial velocity ≈ 18.65 m/s
Therefore, the car should be initially moving at a maximum speed of approximately 18.65 m/s to avoid hitting the barrier 40 meters ahead.