A truck is traveling at a constant speed of 20 m/s through a school zone. At time t = 0 seconds, he passes a hidden police car that is at rest. Five seconds after the truck passes, the police car begins accelerating at a constant rate of 2 m/s2 in order to catch the truck.
-
Well, it sounds like the police car wants to make a "speedy" catch! Let's see if it can pull it off!
Since the truck is traveling at a constant speed of 20 m/s, we can say its distance from the police car is increasing at a rate of 20 m/s. After 5 seconds, the police car decides it's time to hit the gas and starts accelerating.
Now, to calculate how long it takes for the police car to catch up with the truck, we need to find the time, let's call it t, when the distance traveled by the police car is equal to the distance traveled by the truck since it passed.
The distance traveled by the police car can be calculated using the equation: distance = initial velocity * time + (1/2) * acceleration * time^2. In this case, the initial velocity of the police car is 0 m/s and the acceleration is 2 m/s^2.
The distance traveled by the truck is simply the speed times the initial time since it passed, which is 20 m/s * 5 s.
So, we have the equation for the police car's distance as 0.5 * 2 m/s^2 * t^2 = 20 m/s * 5 s.
Now, let's solve for t: 2 * t^2 = 100.
Dividing by 2, we get: t^2 = 50.
And finally, taking the square root of both sides, we get: t ≈ 7.07 seconds.
So, according to my calculations, it will take approximately 7.07 seconds for the police car to catch up with the speeding truck. I hope they both enjoy the chase!
To find out when the police car catches up to the truck, we need to determine the distance each vehicle travels and set them equal to each other.
Let's first calculate the distance the truck travels in the first 5 seconds.
Distance = Speed × Time
Distance = 20 m/s × 5 s
Distance = 100 m
Next, let's find the equation for the distance traveled by the police car.
The equation for distance of an object undergoing constant acceleration is given by:
Distance = Initial velocity × Time + 0.5 × Acceleration × Time^2
The initial velocity of the police car is 0 m/s, and the acceleration is 2 m/s^2.
So the equation for the distance traveled by the police car after 5 seconds is:
Distance = 0.5 × 2 m/s^2 × (Time - 5 s)^2
Now, let's set the distance of the truck equal to the distance of the police car and solve for time:
100 m = 0.5 × 2 m/s^2 × (Time - 5 s)^2
Multiply both sides by 2 m/s^2 to get rid of the 0.5 factor:
200 m/s^2 = (Time - 5 s)^2
Take the square root of both sides:
√(200 m/s^2) = Time - 5 s
Simplifying the square root:
√200 = Time - 5 s
Adding 5 s to both sides:
Time = 5 s + √200
Therefore, the police car catches up to the truck after 5 seconds plus the square root of 200 seconds.
To solve this problem, we need to determine the time it will take for the police car to catch up with the truck.
First, let's find the position of the truck as a function of time. Since the truck is traveling at a constant speed, we can use the formula:
position_of_truck = initial_position + (speed_of_truck * time)
Given that the truck passes the police car at t = 0 seconds and is traveling at a constant speed of 20 m/s, we can plug in these values:
position_of_truck = (20 m/s) * time
Now, let's find the position of the police car as a function of time. Since the police car starts from rest and accelerates at a constant rate, we can use the formula:
position_of_police_car = initial_position + (initial_velocity * time) + (0.5 * acceleration * time^2)
Given that the police car starts moving 5 seconds after the truck passes and accelerates at a rate of 2 m/s^2, we can plug in these values:
position_of_police_car = 0 + (0 * time) + (0.5 * 2 m/s^2 * (time - 5 seconds)^2)
Now, at the time when the police car catches up with the truck, their positions will be equal. Therefore, we can set the equations for the positions of the truck and police car equal to each other:
(20 m/s) * time = 0 + (0 * time) + (0.5 * 2 m/s^2 * (time - 5 seconds)^2)
Now, we can solve this equation for time. Simplifying the equation, we get:
20 m/s * time = 2 m/s^2 * 0.5 * (time - 5 seconds)^2
20 m/s * time = 1 m/s^2 * (time - 5 seconds)^2
20*time = (time - 5)^2
Expanding the right side, we get:
20*time = time^2 - 10*time + 25
Now, rearranging the equation, we get:
time^2 - 30*time + 25 = 0
This is a quadratic equation. We can solve it by factoring, completing the square, or using the quadratic formula.
Once we solve for time, we will have the time it takes for the police car to catch up with the truck.
I expect he'll catch up.
truck: s = 20t
cop: s = 2/2 (t-5)^2
Now you can find how long or how far it takes the cop to catch up.