Two bank robbers 1are in their get-way car traveling at it's top speed of 30m/s. They drive by a parked pollice officer who was on the lookout for them. If the police officer takes 2 seconds to react and engage in pursuit , how long will it take him to catch the robbers if his carhas a maximum acceleration of 4.5m/s^2. You assume the police officers car has a greater top speed than the robbers car
Well, well, well, looks like we've got a high-speed chase on our hands! Let's crunch those numbers.
First, we need to find out how far the robbers travel in those 2 seconds. Since they're driving at a constant speed of 30 m/s, we can use the formula distance = speed × time. So, distance = 30 m/s × 2 s, which gives us 60 meters.
Now, the police officer needs to catch up to that distance in order to engage in pursuit. To figure out how long that'll take, we can use the equation final velocity = initial velocity + acceleration × time.
Since the police officer's car has a maximum acceleration of 4.5 m/s^2, we'll plug in the values: final velocity = 30 m/s, initial velocity = 0 m/s, and we already know the distance traveled is 60 meters. Rearranging the equation, we get time = (final velocity - initial velocity) / acceleration.
So, time = (30 m/s - 0 m/s) / 4.5 m/s^2, which gives us 6.67 seconds.
And there you have it! It will take the police officer approximately 6.67 seconds to catch up to those pesky bank robbers. Safe driving, officer!
To determine how long it will take for the police officer to catch the robbers, we can use the following steps:
Step 1: Determine the initial distance between the police officer and the robbers.
Since the police officer was parked and the robbers were traveling at a constant speed, the initial distance is equal to the speed of the robbers multiplied by the reaction time of the police officer:
Initial distance = 30 m/s * 2 s = 60 m
Step 2: Determine the maximum speed of the police officer's car.
We are given that the police officer's car has a greater top speed than the robbers' car, so let's assume the police officer's car has a top speed of 40 m/s.
Step 3: Determine the time required for the police officer to accelerate to the maximum speed.
Using the equation for acceleration:
acceleration = change in velocity / time
4.5 m/s^2 = (40 m/s - 0 m/s) / time
time = (40 m/s - 0 m/s) / 4.5 m/s^2
time = 8.89 s
Step 4: Determine the distance the police officer can cover during the acceleration phase.
Using the equation for distance traveled during acceleration:
distance = (initial velocity * time) + (0.5 * acceleration * time^2)
distance = (0 m/s * 8.89 s) + (0.5 * 4.5 m/s^2 * (8.89 s)^2)
distance ≈ 177.21 m
Step 5: Determine the remaining distance between the police officer and the robbers.
Remaining distance = Initial distance - Distance covered during acceleration
Remaining distance = 60 m - 177.21 m
Remaining distance ≈ -117.21 m
Since the remaining distance is negative, it means the police officer has already caught up to the robbers during the acceleration phase. Therefore, it will take approximately 8.89 seconds for the police officer to catch the robbers.
To find out how long it will take for the police officer to catch the robbers, we need to determine the distance the robbers will travel before the police officer catches up to them.
First, we calculate the distance the robbers will travel during the 2 seconds it takes the police officer to react:
Distance = Initial Velocity * Time + (1/2) * Acceleration * Time^2
In this case, the initial velocity of the robbers is 30 m/s, the time is 2 seconds, and since the robbers are already at their top speed, their acceleration is 0.
Distance = 30 * 2 + (1/2) * 0 * 2^2
Distance = 60 meters
Now, we need to determine the time it takes for the police officer to cover this distance with his maximum acceleration. We can use the equation of motion:
Distance = Initial Velocity * Time + (1/2) * Acceleration * Time^2
Here, the initial velocity of the police officer is 0 m/s (since he starts from rest), the distance is 60 meters, and the acceleration is 4.5 m/s^2.
60 = 0 * T + (1/2) * 4.5 * T^2
We can simplify this equation to:
4.5 * T^2 = 60
Solving for T, we divide both sides of the equation by 4.5:
T^2 = 60 / 4.5
T^2 = 13.3333...
Taking the square root of both sides, we find:
T ≈ √13.3333...
T ≈ 3.65 seconds
Therefore, it will take approximately 3.65 seconds for the police officer to catch the robbers.
Of course you have to assume the cops can go faster than the robbers -- how else could they ever catch them?
t seconds after passing the parked car,
the robbers have traveled 30t meters
The cops have gone 1/2 * 4.5 * (t-2)^2
So, just solve
30t = 2.25 (t-2)^2
d1 = 30m/s * 2s = 60 m.
60 + 0.5a*t^2 = 30t.
2.25t^2 - 30t + 60 = 0,