A motorcycle officer hidden at an intersection observes a car driven by an oblivious driver who ignores a stop sign and continues through the intersection at constant speed. The police officer takes off in pursuit 1.70 s after the car has passed the stop sign. She accelerates at 4.2 m/s2 until her speed is 104 km/h, and then continues at this speed until she catches the car. At that instant, the car is 1.2 km from the intesection.
To solve this problem, we need to find the time it takes for the motorcycle officer to catch up with the car.
Let's break down the problem into several steps:
Step 1: Convert the speed of the officer to m/s.
The speed of the officer is given as 104 km/h. To convert it to m/s, we multiply it by (1000 m / 1 km) and divide by (60 s / 1 min) and (60 min / 1 hour):
Speed of officer = (104 km/h) x (1000 m / 1 km) x (1 hour / 3600 s) ≈ 28.9 m/s
Step 2: Find the time it took for the officer to reach 104 km/h.
The officer accelerates at 4.2 m/s^2 until she reaches 104 km/h. We can use the following kinematic equation to find the time it took:
Vf = Vi + at
where Vf is the final velocity, Vi is the initial velocity, a is the acceleration, and t is the time.
Since the initial velocity (Vi) is 0 m/s, and the final velocity (Vf) is 28.9 m/s, we can rearrange the equation to solve for t:
t = (Vf - Vi) / a
t = (28.9 m/s - 0 m/s) / 4.2 m/s^2
t ≈ 6.9 seconds
Step 3: Find the distance the officer traveled during acceleration.
To find the distance the officer traveled during acceleration, we use the following kinematic equation:
d = Vit + 0.5at^2
where d is the distance, Vi is the initial velocity, a is the acceleration, and t is the time.
Since the initial velocity (Vi) is 0 m/s, the acceleration (a) is 4.2 m/s^2, and the time (t) is 6.9 seconds, we can substitute these values into the equation to calculate the distance:
d = (0 m/s)(6.9 s) + 0.5(4.2 m/s^2)(6.9 s)^2
d ≈ 83.6 meters
Step 4: Find the time it takes for the officer to catch the car.
The total distance between the intersection and the car is given as 1.2 km, which is equal to 1200 meters. Since the officer starts 83.6 meters behind the car, the effective distance she needs to cover is:
Effective distance = Total distance - Distance during acceleration
Effective distance = 1200 m - 83.6 m
Effective distance = 1116.4 m
To find the time it takes to cover this distance, we use the equation:
t = d / v
where d is the distance and v is the velocity.
t = 1116.4 m / 28.9 m/s
t ≈ 38.6 seconds
Therefore, it takes approximately 38.6 seconds for the officer to catch the car after accelerating.
To solve this problem, we can break it down into three key parts:
1. Determining the time it takes for the police officer to reach a speed of 104 km/h.
2. Calculating the distance traveled by the police officer from the intersection until she reaches 104 km/h.
3. Finding the time it takes for the police officer to catch up to the car.
Let's go through each of these steps:
1. Determining the time it takes for the police officer to reach a speed of 104 km/h:
To find this, we can use the following equation:
v = u + at
Where:
v = final velocity (104 km/h)
u = initial velocity (0 km/h, as the officer starts from rest)
a = acceleration (4.2 m/s²)
t = time
First, we need to convert the final velocity from km/h to m/s:
104 km/h * (1000 m/1 km) * (1/3600 h/1 s) = 28.9 m/s
Now we can rearrange the equation to solve for time (t):
t = (v - u) / a
t = (28.9 m/s - 0 m/s) / (4.2 m/s²)
t ≈ 6.88 s
Thus, it takes approximately 6.88 seconds for the police officer to reach a speed of 104 km/h.
2. Calculating the distance traveled by the police officer from the intersection until she reaches 104 km/h:
To find this, we can use the following equation:
s = ut + (1/2)at²
Where:
s = distance traveled
u = initial velocity (0 m/s)
t = time (6.88 s)
a = acceleration (4.2 m/s²)
Plugging the values into the equation:
s = (0 m/s)(6.88 s) + (1/2)(4.2 m/s²)(6.88 s)²
s ≈ 83.8 m
Thus, the police officer travels approximately 83.8 meters from the intersection until she reaches a speed of 104 km/h.
3. Finding the time it takes for the police officer to catch up to the car:
To find this, we need to determine the relative distance between the police officer and the car at any given time. When they eventually meet, this distance will be zero.
Relative distance = Distance traveled by the car - Distance traveled by the police officer
Given that the car is 1.2 km (1200 m) away from the intersection, we subtract the distance traveled by the police officer (83.8 m) to get the distance between them at the start of the pursuit:
Relative distance = 1200 m - 83.8 m
Relative distance = 1116.2 m
Now, we can calculate the time it takes for the police officer to catch up to the car using the equation:
Relative distance = Car's velocity * Time
Since the car is moving at a constant speed, we can write this equation as:
1116.2 m = Car's speed * Time
Rearranging the equation to solve for time (T):
Time (T) = Relative distance / Car's speed
Let's convert the car's speed to m/s:
104 km/h * (1000 m/1 km) * (1/3600 h/1 s) = 28.9 m/s
T = 1116.2 m / 28.9 m/s
T ≈ 38.6 s
Thus, it takes approximately 38.6 seconds for the police officer to catch up to the car.
In summary, the police officer takes about 6.88 seconds to reach a speed of 104 km/h, travels approximately 83.8 meters from the intersection to reach that speed, and catches up to the car in approximately 38.6 seconds.