A car is travelling along a highway with a speed of 25 m/s when the driver sees an obstruction 1.80*10^2 m directly ahead. It takes the driver 0.80s to react and begin braking.
a) How far does the car travel before it begins to slow down?
b) How long will it take the car to stop once the brakes are applied, provided the car stops just before the obstruction?
c) What is the value of the acceleration of the car if it just misses hitting the obstruction? Assume the acceleration is uniform.
a) How far does the car travel before it begins to slow down?
Well, before we calculate that, let's hope the driver doesn't panic and hit the clown car's brake button instead of the car's brake pedal! Okay, let's get serious now.
To calculate the distance traveled before the car begins to slow down, we need to find the distance covered during the reaction time of 0.80s. Since the car is already moving with a speed of 25 m/s, we can use the formula:
Distance = Speed × Time
So, Distance = (25 m/s) × (0.80s)
I'll leave it to you to calculate the result. Just promise me you won't be clowning around while doing it!
To answer these questions, we need to use the basic equations of motion.
Let's start with part a:
a) How far does the car travel before it begins to slow down?
To calculate the distance the car travels before it begins to slow down, we can use the equation:
distance = initial velocity * reaction time
distance = 25 m/s * 0.80 s
distance = 20 m
Therefore, the car travels 20 meters before it begins to slow down.
Now, let's move on to part b:
b) How long will it take the car to stop once the brakes are applied, provided the car stops just before the obstruction?
To calculate the time it takes for the car to stop once the brakes are applied, we can use the equation of motion:
final velocity^2 = initial velocity^2 + 2 * acceleration * distance
We know the final velocity is zero (since the car stops) and the initial velocity is 25 m/s. The distance can be calculated as the total distance (1.80 * 10^2 m) minus the distance traveled before braking (20 m).
0 = (25 m/s)^2 + 2 * acceleration * (1.80 * 10^2 m - 20 m)
Simplifying:
0 = 625 m^2/s^2 + 2 * acceleration * 160 m
Rearranging the equation:
acceleration = -625 m^2/s^2 / (2 * 160 m)
acceleration = -1.95 m/s^2
Now that we have the acceleration, let's find the time it takes for the car to stop. We can use the equation of motion:
final velocity = initial velocity + acceleration * time
Since the final velocity is zero, we can rearrange the equation:
time = -initial velocity / acceleration
time = -25 m/s / -1.95 m/s^2
time ≈ 12.82 s
Therefore, it will take approximately 12.82 seconds for the car to stop once the brakes are applied.
Finally, let's move on to part c:
c) What is the value of the acceleration of the car if it just misses hitting the obstruction? Assume the acceleration is uniform.
Since the car just misses hitting the obstruction, the final position of the car is the initial position plus the total distance traveled before braking. Therefore, the final position will be:
final position = initial position + distance
final position = 0 m + 1.80 * 10^2 m
final position = 1.80 * 10^2 m
Using the equation of motion:
final velocity^2 = initial velocity^2 + 2 * acceleration * distance
We can substitute the values:
0 = (25 m/s)^2 + 2 * acceleration * (1.80 * 10^2 m)
Simplifying:
0 = 625 m^2/s^2 + 2 * acceleration * 180 m
Rearranging the equation:
acceleration = -625 m^2/s^2 / (2 * 180 m)
acceleration = -1.73 m/s^2
Therefore, the value of the acceleration of the car, assuming it just misses hitting the obstruction, is approximately -1.73 m/s^2.
To solve these problems, we need to use the equations of motion. Let's go step by step:
a) The car travels before it begins to slow down can be calculated using the equation: distance = initial velocity × time. In this case, the initial velocity is 25 m/s and the time is the reaction time of the driver, which is 0.80s. Therefore, the distance can be calculated as follows:
distance = 25 m/s × 0.80s = 20 meters
Therefore, the car travels 20 meters before it starts to slow down.
b) To calculate the time it takes for the car to stop once the brakes are applied, we can use the equation of motion: distance = initial velocity × time + 0.5 × acceleration × time^2. In this case, the distance is the remaining distance to the obstruction, which is given as 1.80 × 10^2 m. The initial velocity is 25 m/s, and we need to calculate the time it takes for the car to stop (let's call it t) and the acceleration (let's call it a). The equation can be rewritten as:
1.80 × 10^2 m = 25 m/s × t + 0.5 × a × t^2
Since we don't know the acceleration, we need another equation to solve the problem.
c) To find the acceleration of the car if it just misses hitting the obstruction, we can use the equation: final velocity^2 = initial velocity^2 + 2 × acceleration × distance. In this case, the final velocity is 0 m/s (since the car stops just before the obstruction), the initial velocity is 25 m/s, and the distance is the remaining distance to the obstruction, which is given as 1.80 × 10^2 m. The equation can be rewritten as:
0 = (25 m/s)^2 + 2 × a × (1.80 × 10^2 m)
Now, we have two equations (from b and c) with two unknowns (a and t), which can be solved simultaneously to find the values of a and t.