A car travelling at a constant speed of 136 km/hr passes a tropper hidden behind a billboard. One second after the speeding car passes the billboard, the tropper sets in a chase after the car wit ha constant acceleration of 3.8 m/s^2. How far does the trooper travel before he overtakes the speeding car?
The distance traveled is the same.
distancespeeder: velocity*time change 136km/hr to m/s
distancetrooper: 1/2 3.8 (t-1)^2
set them equal, solve for t. then, go back and solve for distance in either equation
The answer above is WRONG.
136 km/h = 38.3333 m/s
x = x0 + vi*t + .5*a*t^2
xtropper = 0 + 0 + .5*3.8*t^2
xspeeder = 0 + 38.3333t + 0
xtropper = xspeeder
.5*3.8*t^2=38.3333t
t = 20.17542105
xtropper = .5*3.8*20.17542105^2
xtropper = 773.3904676
To find the distance the trooper travels before overtaking the car, we need to find the time it takes for the trooper to catch up to the car.
Let's start by converting the car's speed from km/hr to m/s.
Given:
Car's speed = 136 km/hr
Trooper's acceleration = 3.8 m/s^2
Converting the car's speed:
136 km/hr * (1000 m / 1 km) * (1 hr / 3600 s) = 37.78 m/s
Now, let's find the time it takes for the trooper to catch up to the car. We can use the equation:
distance = initial velocity * time + (1/2) * acceleration * time^2
For the car, the initial velocity is 37.78 m/s and the acceleration is 0 m/s^2 (since it's traveling at a constant speed). Therefore:
distance_car = 37.78 m/s * t
For the trooper, the initial velocity is 0 m/s (since it starts from rest), and the acceleration is 3.8 m/s^2. Therefore:
distance_trooper = (1/2) * 3.8 m/s^2 * t^2
Since we want to find the time it takes for the trooper to catch up to the car, we can set the distances equal to each other:
37.78 m/s * t = (1/2) * 3.8 m/s^2 * t^2
Simplifying the equation:
37.78 * t = 1.9 * t^2
Rearranging the equation:
1.9 * t^2 - 37.78 * t = 0
Now, we can solve this quadratic equation to find the value of t:
1.9 * t^2 - 37.78 * t = 0
This equation can be factored as:
t * (1.9 * t - 37.78) = 0
The solutions for t are:
t = 0 (Ignoring this since we are looking for a positive time value to represent the trooper overtaking the car)
t = 37.78 / 1.9
t ≈ 19.88 s
Therefore, it takes approximately 19.88 seconds for the trooper to catch up to the car.
To find the distance traveled by the trooper before overtaking, we can substitute this time (t) into the distance formula for the trooper:
distance_trooper = (1/2) * 3.8 m/s^2 * (19.88 s)^2
Calculating the distance:
distance_trooper ≈ (1/2) * 3.8 m/s^2 * (395.21 s^2)
distance_trooper ≈ 3744.39 meters
Therefore, the trooper travels approximately 3744.39 meters before overtaking the speeding car.
To find the distance the trooper travels before overtaking the speeding car, we need to determine the time it takes for the trooper to catch up with the car.
First, let's convert the speed of the car to meters per second (m/s):
Car's speed = 136 km/hr = (136 * 1000) m / (3600 s) ≈ 37.78 m/s
Now, let's find the time it takes for the trooper to catch up with the car. We can use the equation of motion:
s = ut + (1/2)at^2
where s is the distance traveled, u is the initial velocity, a is the acceleration, and t is the time.
For the car, initial velocity u = 37.78 m/s and acceleration a = 0 m/s^2 since it is traveling at a constant speed. We'll assume that the trooper starts from rest, so his initial velocity u = 0 m/s.
Let's rearrange the equation to solve for time (t):
s = ut + (1/2)at^2
0 = (1/2)at^2 + ut - s
(1/2)at^2 + ut - s = 0
Now we can use the quadratic formula to solve for t:
t = (-u ± √(u^2 - 4(1/2)a(-s))) / (2(1/2)a)
t = (-u ± √(u^2 + 2as)) / a
Substituting the values:
t = (-0 ± √((0)^2 + 2 * 3.8 * s)) / 3.8
t = √(2s) / 3.8
Since the trooper starts chasing one second after the car passes, the total time for the trooper to catch up is t + 1.
Now, we can substitute the value of t in terms of s into the equation:
t + 1 = √(2s) / 3.8 + 1
To find the value of s, we need to solve this equation. We'll start by subtracting 1 from both sides:
t = √(2s) / 3.8
Now, let's square both sides of the equation to eliminate the square root:
t^2 = 2s / (3.8)^2
s = (t^2 * (3.8)^2) / 2
Finally, substitute the value of t, which is the time to travel, into the equation to find s:
s = (t^2 * (3.8)^2) / 2
Now you can calculate the distance the trooper travels before overtaking the car by plugging in the value of time (t) into this equation and solving it!