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 1.56 s after the car has passed the stop sign. She accelerates at 3.6 m/s2 until her speed is 115 km/h, and then continues at this speed until she catches the car. At that instant, the car is 1.0 km from the intesection.
(a) How long did it take for the officer to catch up to the car? (in seconds)
(b) How fast was the car traveling? (in mi/h)
To solve this problem, we need to analyze the motion of both the police officer and the car. Let's break it down step by step:
Step 1: Calculate the time it took for the police officer to reach the speed of 115 km/h.
To do this, we can use the equation of motion: final velocity = initial velocity + acceleration * time.
We know the initial velocity is 0 m/s since the officer was stationary when she started accelerating. The final velocity is 115 km/h, which we need to convert to m/s.
115 km/h = 115000 m/3600 s = 31.94 m/s.
Using the equation of motion,
31.94 m/s = 0 m/s + 3.6 m/s^2 * t.
Solving for t, we find t = 31.94 m/s / (3.6 m/s^2) ≈ 8.87 s.
So it took approximately 8.87 seconds for the officer to reach a speed of 115 km/h.
Step 2: Calculate the distance the officer traveled during the acceleration phase.
We can use the equation of motion: distance = initial velocity * time + (1/2) * acceleration * time^2.
Here, initial velocity = 0 m/s, time = 8.87 s, and acceleration = 3.6 m/s^2.
Plugging the values into the equation, we get:
distance = 0 m/s * 8.87 s + (1/2) * 3.6 m/s^2 * (8.87 s)^2.
Simplifying, we find:
distance = 0 m + 0.5 * 3.6 m/s^2 * 78.69 s^2.
distance ≈ 141.65 m.
So the officer traveled approximately 141.65 meters during the acceleration phase.
Step 3: Calculate the time it takes for the officer to catch the car.
Let's assume the time taken by the officer to catch the car is "t" seconds after she starts accelerating. Since the car has been moving at a constant speed, it has traveled a distance of 1.0 km (1000 m) in that time.
So the total distance traveled by the officer = distance during acceleration phase + distance traveled at a constant speed = 141.65 m + 1000 m = 1141.65 m.
Using the equation of motion, distance = velocity * time, we can find the time taken to catch the car:
1141.65 m = 31.94 m/s * t.
Solving for t, we find t = 1141.65 m / 31.94 m/s ≈ 35.71 s.
So, it took approximately 35.71 seconds for the officer to catch up to the car.
Step 4: Calculate the speed of the car.
During the time taken for the officer to catch the car, the car traveled a distance of 1.0 km (1000 m). Divide this distance by the time taken to find the speed of the car:
Speed of the car = distance / time = 1000 m / 35.71 s.
Converting this to miles per hour (mi/h), we have:
Speed of the car (mi/h) = (1000 m / 35.71 s) * (1 km / 1000 m) * (1 mi / 1.61 km) * (3600 s / 1 h).
Simplifying, we find:
Speed of the car (mi/h) ≈ 79.96 mi/h.
So the car was traveling at approximately 79.96 miles per hour before being caught by the police officer.
(a) The officer took approximately 35.71 seconds to catch up to the car.
(b) The car was traveling at approximately 79.96 miles per hour.