A car traveling at a constant speed of 125 km/hr passes a trooper hidden behind a billboard. One second after the speeding car passes the billboard, the trooper sets in a chase after the car with a constant acceleration of 2.7 m/s^2. How far does the trooper travel before he overtakes the speeding car?
I'm pretty sure that I need to solve for time using d = 1/2at^2, but the "one second after" part of this problem throws me off.
125 km/hr *1 hr/3600 s * 1000m/km =
34.7 m/s
the car travels t+1 seconds
so the car goes
d =34.7 * (t+1) = 34.7 t + 34.7 meters
the trooper travels the same distance in t seconds
d = 0 + 0t + (1/2)(2.7)t^2
so
34.7 t + 34.7 = 1.35 t^2
1.35 t^2 - 34.7 t - 34.7 = 0
solve quadratic, use solution that makes sense.
Thanks. :)
To solve this problem, you are correct in using the kinematic equation for distance:
d = 1/2 * a * t^2
Let's break down the problem step by step:
Step 1: Convert the speed of the car from km/h to m/s.
Given: Car's constant speed = 125 km/hr
To convert km/hr to m/s, divide by 3.6 (1 km/hr = 1/3.6 m/s).
Car's constant speed = 125 km/hr * (1/3.6) m/s = 34.72 m/s (approximately)
Step 2: Determine the time taken by the car to cross the billboard.
Given: The car passes the billboard and one second later, the trooper starts the chase.
Time taken by the car to cross the billboard = 1 second
Step 3: Calculate the time for the trooper to overtake the speeding car.
Given: Trooper's acceleration = 2.7 m/s^2
We can use the equation of motion:
s = ut + 1/2at^2
Where s is the displacement, u is the initial velocity, a is the acceleration, and t is the time.
Let's consider the trooper's initial velocity, u, as 0 m/s since they started chasing the car from rest.
Using this equation, we can rearrange it to solve for time, t:
s = 1/2at^2
2s = at^2
t^2 = 2s / a
t = sqrt(2s / a)
Now, we need to find the displacement, s, which is the distance traveled by the car during the time taken by the trooper.
Given: the initial velocity of the car, u = 34.72 m/s
Time taken by the trooper, t = 1 second
Acceleration of the trooper, a = 2.7 m/s^2
Substituting these values into the equation, we have:
t = sqrt(2s / a)
1 = sqrt(2u / a)
1 = sqrt(2 * 34.72 / 2.7)
1 = sqrt(2 * 12.87)
1 = sqrt(25.74)
1 = 5.07 (approximately)
So, the time taken by the trooper to catch up to the speeding car is approximately 5.07 seconds.
Step 4: Calculate the distance traveled by the trooper to catch up to the car.
Using the equation of motion:
s = ut + 1/2at^2
Given: u = 0 m/s (initial velocity of the trooper)
t = 5.07 seconds (time calculated in Step 3)
a = 2.7 m/s^2 (acceleration of the trooper)
Substituting these values into the equation, we have:
s = ut + 1/2at^2
s = 0 * 5.07 + 1/2 * 2.7 * 5.07^2
s = 0 + 0.5 * 2.7 * 25.70
s = 34.53 meters (approximately)
Therefore, the trooper travels approximately 34.53 meters before overtaking the speeding car.
To solve this problem, you are on the right track by using the equation d = (1/2)at^2. However, since you are given the initial velocity of the car, you can use the equation to find the time it takes for the trooper to catch up to the car.
First, convert the car's speed from km/hr to m/s by dividing by 3.6:
Initial velocity of the car = 125 km/hr = (125 * 1000) m/3600 s = 34.72 m/s
Next, let's find the time it takes for the trooper to catch up to the car. We can use the equation v = u + at, where:
v = final velocity (both the car and the trooper will have the same velocity when the trooper catches up)
u = initial velocity (velocity of the car)
a = acceleration of the trooper
t = time
Since the trooper starts from rest (initial velocity is 0 m/s), the equation simplifies to v = at.
Substituting the given values, we have:
34.72 m/s = 2.7 m/s^2 * t
Now solve for t:
t = 34.72 m/s / 2.7 m/s^2
t = 12.86 s
Therefore, it takes approximately 12.86 seconds for the trooper to catch up to the car.
Now, to find the distance traveled by the trooper before catching up with the car, we can use the equation d = ut + (1/2)at^2.
Substituting the values into the equation, we get:
d = 0 m/s * 12.86 s + (1/2) * 2.7 m/s^2 * (12.86 s)^2
Calculating the equation, we find:
d = 0 + 0.5 * 2.7 * (165.4)
d = 224.62 meters
Hence, the trooper travels approximately 224.62 meters before overtaking the speeding car.