A speeder passes a parked police car at a
constant speed of 20.1 m/s. At that instant,
the police car starts from rest with a uniform
acceleration of 2.65 m/s
2
.
How much time passes before the speeder
is overtaken by the police car?
Answer in units of s.
the distance traveled is the same, in the same time.
dspeeder=20.1*t
dscop=1/2 *2.65 t^2
set them equal, and solve for time t.
To find the time it takes for the police car to overtake the speeder, we need to determine the time it would take for both vehicles to reach the same position.
Let's first calculate the distance traveled by each vehicle:
The speeder travels at a constant speed of 20.1 m/s. Let's denote the distance traveled by the speeder as "D".
The police car starts from rest and undergoes uniform acceleration. We'll denote the distance traveled by the police car as "d".
The distance traveled by an object can be calculated using the equation:
distance = initial velocity * time + 0.5 * acceleration * time^2
For the speeder:
D = 20.1 m/s * t (since it travels at a constant speed)
For the police car:
d = 0.5 * 2.65 m/s^2 * t^2
Now, we can set the two distances equal to each other to find the time it takes for the police car to overtake the speeder:
20.1 m/s * t = 0.5 * 2.65 m/s^2 * t^2
We simplify this equation:
20.1 m/s * t = 1.325 m/s^2 * t^2
Rearranging the equation:
1.325 m/s^2 * t^2 - 20.1 m/s * t = 0
This is a quadratic equation in terms of "t". We can solve this equation using the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / 2a
In this case, we have:
a = 1.325 m/s^2
b = -20.1 m/s
c = 0
Applying these values to the quadratic formula:
t = (-(-20.1) ± √((-20.1)^2 - 4 * 1.325 * 0)) / (2 * 1.325)
Simplifying further:
t = (20.1 ± √(404.01)) / 2.65
Calculating the square root of 404.01:
t = (20.1 ± 20.1) / 2.65
Now we have two possible values for "t":
t₁ = (20.1 + 20.1) / 2.65 ≈ 15.0943 s
t₂ = (20.1 - 20.1) / 2.65 ≈ 0 s
The time cannot be negative, so t₂ = 0 s is not valid. Therefore, the time it takes for the police car to overtake the speeder is approximately t = 15.0943 s.