A driver of a car going 90.0km/h suddenly sees the lights of a barrier 40.0 metres ahead. It takes the driver 0.75 seconds before she applies the brakes, and the average acceleration during braking is -10.0 m/s squared. Does the car hit the barrier?SHow all your work. (2 parts)
V = 90km/h = 90km/h / 3600s/h = 0.025km/s = 25m/s.
d = 25m/s * 0.75s = 18.75m = Distance
traveled before brakes are applied.
d = 40 - 18.75 = 21.25m = max stopping
dist.
Vf^2 = Vo^2 + 2ad = 0,
(25)^2 + 2(-10)d = 0,
625 - -20d = 0,
-20d = -625,
d = -625 / -20 = 31.25m = Distance traveled after brakes are applied.
The car will hit the barrier, because it was required to stop within 21.25m
after the brakes are applied.
Well, let's do the math and find out if the car is going to embrace the barrier or narrowly avoid it!
First, we need to calculate the initial velocity of the car before braking:
Initial velocity, u = 90.0 km/h = 90.0 * (1000/3600) m/s
≈ 25.0 m/s
Next, let's find out how much distance the car covers before the driver applies the brakes:
Distance covered before applying brakes, d = (Initial velocity * Time) + (1/2 * Acceleration * Time²)
= (25.0 * 0.75) + (1/2 * (-10.0) * (0.75)²)
≈ 18.8 meters
So, before applying the brakes, the car travels approximately 18.8 meters.
Now, let's calculate how much distance the car needs to stop completely:
Stopping distance, s = (Initial velocity²) / (2 * Absolute value of Acceleration)
= (25.0²) / (2 * 10.0)
≈ 31.3 meters
The car needs approximately 31.3 meters to stop completely.
Finally, we can determine if the car hits the barrier or not by comparing the distance covered before applying the brakes with the stopping distance:
Distance covered before applying brakes < Stopping distance?
18.8 meters < 31.3 meters?
Yes, the car covers less distance before applying the brakes than it needs to stop completely. Therefore, the car does hit the barrier!
To determine if the car hits the barrier, we need to calculate the distance it travels before coming to a stop.
First, let's convert the speed from km/h to m/s:
90.0 km/h = (90.0 × 1000) m/ (3600 s) = 25 m/s
Part 1: Calculating the distance traveled before applying the brakes
Since it takes the driver 0.75 seconds before applying the brakes, the car will continue moving at the initial speed for this time:
Distance = Speed × Time
Distance = 25 m/s × 0.75 s
Distance = 18.75 meters
Part 2: Calculating the distance traveled during braking
To find the distance traveled during braking, we can use the equation of motion:
Distance = (Initial Speed × Time) + (1/2) × Acceleration × (Time^2)
Here, the initial speed is 25 m/s, the time taken for braking is 0.75 s, and the acceleration is -10.0 m/s^2. Plugging in these values, we get:
Distance = (25 m/s × 0.75 s) + (1/2) × (-10.0 m/s^2) × (0.75 s)^2
Simplifying the equation, we have:
Distance = 18.75 m + (-3.75 m)
Distance = 15 meters
Therefore, the total distance traveled by the car before coming to a stop is:
Total Distance = Distance during acceleration + Distance during braking
Total Distance = 18.75 meters + 15 meters
Total Distance = 33.75 meters
Since the total distance the car travels (33.75 meters) is less than the distance to the barrier (40.0 meters), the car will hit the barrier.
To determine whether the car hits the barrier, we need to calculate the distance the car will travel during the reaction time of 0.75 seconds and the distance it will take to stop completely.
First, let's find the distance the car travels during the reaction time. We will use the formula:
d = vt
Where:
d = distance
v = velocity
t = time
Given that the velocity (v) is 90.0 km/h and the time (t) is 0.75 seconds, we need to convert the velocity to meters per second (m/s):
90.0 km/h * (1000 m/km) * (1 h/3600 s) = 25.0 m/s
Now, we can plug our values into the formula:
d = 25.0 m/s * 0.75 s = 18.75 meters
Therefore, the car travels 18.75 meters during the reaction time.
Secondly, let's find the distance it takes for the car to stop completely. We will use the following formula of motion:
v^2 = u^2 + 2as
Where:
v = final velocity (0 m/s, as the car stops)
u = initial velocity (25.0 m/s, as calculated earlier)
a = acceleration (-10.0 m/s^2, deceleration in this case)
s = distance
Rearranging the formula, we have:
s = (v^2 - u^2) / (2a)
Plugging in the values, we get:
s = (0^2 - (25.0 m/s)^2) / (2 * -10.0 m/s^2)
s = -625 / -20
s = 31.25 meters
Therefore, the distance it takes the car to stop completely is 31.25 meters.
Now, let's calculate the total distance traveled by the car during the reaction time and braking:
Total distance = distance during reaction time + distance to stop completely
Total distance = 18.75 meters + 31.25 meters
Total distance = 50 meters
Comparing the total distance (50 meters) with the distance to the barrier (40 meters), we can conclude that the car will hit the barrier since the total distance traveled is greater than the distance to the barrier.