Two cars are travelling at adjacent highway lanes in the same direction . Car B is stopped by car A which travels at 15km/hr. Then 2 seconds later car B starts and accelerate at a constant rate of 3 meter per seconds square . Determine when and where car B will overtake car A
Determine the speed of car B at that time
time for car a is t+2
time for car b is t
distance car a
= (15000 m/3600 s)(t+2)
distance car b = (1/2)(3)t^2
so when does car a travel the same distance as car b?
(150/36)(t+2) = (3/2) t^2
solve quadratic, use the t that makes sense :)
speed of b = 3 t
To determine when and where car B will overtake car A, we need to find the time it takes for car B to catch up with car A, as well as the distance covered by both cars during that time.
1. Let's first find the time it takes for car B to catch up with car A:
- Car B starts moving 2 seconds after car A.
- The relative speed between the two cars is the difference in their velocities.
- The relative speed can be calculated as (15 km/h - 0 km/h) = 15 km/h = (15 * 1000)/(60 * 60) m/s = 4.17 m/s.
- Using the equation s = ut + (1/2)at^2, where s is the distance covered, u is the initial velocity, t is time, and a is acceleration, we can find the time it takes for car B to catch up using the equation s = ut + (1/2)at^2.
- In this case, s is the distance traveled by car B in catching up with car A, u is the initial velocity of car B (0 m/s), t is the time it takes, and a is the acceleration of car B (3 m/s^2).
- Let's solve for t: 4.17 m/s * t + (1/2)(3 m/s^2)t^2 = 0. Using the quadratic formula, we can find the value of t.
2. Once we have the time it takes for car B to catch up with car A, we can find the distance covered by both cars during that time:
- Using the equation s = ut + (1/2)at^2, we can find the distance traveled by car A during the time it took for car B to catch up.
- Car A's initial velocity is 15 km/h, which is (15 * 1000)/(60 * 60) m/s = 4.17 m/s, and the time is the same as the time it took for car B to catch up.
- Let's solve for s.
3. To determine the speed of car B at that time, we need to calculate its velocity at the time when it catches up with car A:
- The velocity of car B at that time would be its initial velocity plus the product of acceleration and time.
- Let's calculate the speed of car B at that time using the equation v = u + at, where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time it took for car B to catch up.
By following these steps, we can determine when and where car B will overtake car A as well as the speed of car B at that time.
To determine when and where Car B will overtake Car A, we need to find the time it takes for Car B to catch up with Car A.
Let's begin by converting the speed of Car A from km/hr to m/s:
Car A's speed = 15 km/hr = (15 * 1000) m/ (60 * 60) s = 4.17 m/s
Now, let's set up the equation of motion for both cars:
For Car A:
Displacement = Speed * Time
For Car B:
Displacement = Initial Velocity * Time + 0.5 * Acceleration * Time^2
Since Car A has already traveled for 2 seconds, let's denote the time it takes for Car B to catch up as T.
For Car A:
Displacement = 4.17 m/s * (T + 2)
For Car B:
Displacement = 0 (since it starts from the same point as Car A)
Setting the two equations equal to each other:
4.17 (T + 2) = 0.5 * 3 * T^2
Simplifying the equation:
4.17T + 8.34 = 1.5T^2
Rearranging the equation to a quadratic form:
1.5T^2 - 4.17T - 8.34 = 0
Now we can use the quadratic formula to solve for T:
T = (-b ± √(b^2 - 4ac)) / 2a
Plugging in the values:
T = (-(-4.17) ± √((-4.17)^2 - 4 * 1.5 * (-8.34))) / (2 * 1.5)
T = (4.17 ± √(17.32 + 19.92)) / 3
T = (4.17 ± √37.24) / 3
Calculating the two possible values of T:
T ≈ 0.434 seconds or T ≈ 4.066 seconds
Since time cannot be negative, we can ignore the negative value of T.
Therefore, Car B will overtake Car A approximately 4.066 seconds after it starts.
To find the speed of Car B at that time, we can use the equation of motion for Car B:
Displacement = Initial Velocity * Time + 0.5 * Acceleration * Time^2
Since Car B starts from rest, the initial velocity (u) is 0 m/s.
Displacement = 0 + 0.5 * 3 * (4.066)^2
Displacement = 0 + 0.5 * 3 * 16.536
Displacement ≈ 24.78 meters
Therefore, Car B will overtake Car A approximately 4.066 seconds after it starts, and the distance between them at that time will be approximately 24.78 meters.
Note: This solution assumes constant acceleration for Car B.