A car moving at constant velocity of 14 m/s passes a gasoline station. two seconds later, another car leaves the gasoline station and accelerates at the constant rate of 2m/s^2. How soon will the second car overtake the first?
To solve this problem, we can set up two separate equations for the distances traveled by each car and find the time at which they are equal.
Let's assume the time it takes for the second car to overtake the first car is "t" seconds.
For the first car:
Distance = Rate × Time
d1 = 14 m/s × t
For the second car:
Since the second car starts moving after two seconds,
Distance = Initial velocity × Time + (1/2) × acceleration × Time²
d2 = 0 + 0 + (1/2) × 2 m/s² × (t - 2)²
To find the time at which the two distances are equal, we'll set the equations d1 and d2 equal to each other and solve for "t":
14t = (1/2) × 2 m/s² × (t - 2)²
Now, let's solve this equation step by step:
Expand the right side of the equation:
14t = (1/2) × 2 m/s² × (t² - 4t + 4)
Multiply:
14t = t² - 4t + 4
Rearrange the equation to bring it to standard quadratic form:
0 = t² - 18t + 4
Now, we can solve this quadratic equation.
Using the quadratic formula, which is:
t = (-b ± √(b² - 4ac)) / 2a
In this equation, a = 1, b = -18, and c = 4.
t = (18 ± √((-18)² - 4(1)(4))) / (2(1))
Calculating inside the square root:
t = (18 ± √(324 - 16)) / 2
t = (18 ± √308) / 2
Now, let's calculate the two possible values for "t":
t₁ = (18 + √308) / 2
t₂ = (18 - √308) / 2
Calculating the square root:
t₁ ≈ 16.573 seconds
t₂ ≈ 1.427 seconds
Since the time cannot be negative, we discard t₂, which gives us the solution:
The second car will overtake the first car approximately 16.573 seconds after it leaves the gasoline station.
To find out how soon the second car will overtake the first car, we need to calculate the time it takes for the second car to catch up to the first car.
Let's start by finding the distance traveled by the first car during the two seconds it took for the second car to start. Since the first car is moving at a constant velocity, we can calculate the distance it travels using the formula:
Distance = Velocity × Time
Distance = 14 m/s × 2 s = 28 m
Now, let's find out how long it will take for the second car to cover this distance and catch up to the first car.
We can use the following kinematic equation to calculate the time it takes for the second car to catch up:
Distance = Initial Velocity × Time + 0.5 × Acceleration × Time^2
Since the initial velocity of the second car is 0 m/s (as it starts from rest), the equation simplifies to:
Distance = 0.5 × Acceleration × Time^2
Plugging in the values we know:
28 m = 0.5 × 2 m/s^2 × Time^2
Divide both sides of the equation by 0.5 × 2 m/s^2:
14 m = Time^2
Taking the square root of both sides:
Time = √(14 m) ≈ 3.74 s
Therefore, the second car will overtake the first car approximately 3.74 seconds after it leaves the gasoline station.