A truck travels at a constant speed of 28.0 m/s in the fast lane of a two-lane highway. It approaches a stationay car stopped at the side of the road. When the truck is still 1.2x10^2 m behind the car, the car pulls out into the slow lane with an acceleration of 2.6 m/s squared.
a)how long will it take the truck to pass the car?
b)how far will the car have travelled when the truck passes it?
c)if the car were to maintain this acceleration, how fast would it be travelling with it overtakes the truck?
someone please answer this, I need this.
To solve this problem, we can use the equations of motion to determine the time it takes for the truck to pass the car, the distance the car will have traveled when the truck passes it, and the speed of the car when it overtakes the truck.
Let's solve each part of the problem step-by-step:
a) To find the time it takes for the truck to pass the car, we can use the equation:
distance = initial velocity * time + 0.5 * acceleration * time^2
Since the car starts from rest, its initial velocity is 0. The distance the truck needs to cover is 1.2x10^2 m, and the car's acceleration is 2.6 m/s^2. Plugging in these values into the equation, we have:
1.2x10^2 = 0 * t + 0.5 * 2.6 * t^2
Rearranging the equation, we get:
1.2x10^2 = 1.3 * t^2
Dividing both sides by 1.3, we have:
t^2 = (1.2x10^2) / 1.3
t^2 ≈ 92.3077
Taking the square root of both sides, we get:
t ≈ √(92.3077)
t ≈ 9.609 seconds
Therefore, it will take the truck approximately 9.609 seconds to pass the car.
b) To find the distance the car will have traveled when the truck passes it, we can use the equation:
distance = initial velocity * time + 0.5 * acceleration * time^2
In this case, the initial velocity of the car is 0, and we already know that it takes approximately 9.609 seconds for the truck to pass the car. We can plug in these values into the equation to find the distance:
distance = 0 * 9.609 + 0.5 * 2.6 * (9.609)^2
distance ≈ 113.3333 meters
Therefore, the car will have traveled approximately 113.3333 meters when the truck passes it.
c) To find the speed of the car when it overtakes the truck, we can use the equation:
final velocity = initial velocity + acceleration * time
Since the car starts from rest, its initial velocity is 0, and we already know that it takes approximately 9.609 seconds for the truck to pass the car. The car's acceleration is 2.6 m/s^2. Plugging in these values into the equation, we have:
final velocity = 0 + 2.6 * 9.609
final velocity ≈ 25.034 m/s
Therefore, if the car were to maintain this acceleration, it would be traveling at approximately 25.034 m/s when it overtakes the truck.
To answer these questions, we need to analyze the motion of both the truck and the car. Let's go step by step:
a) How long will it take the truck to pass the car?
To find this, we need to determine the time it takes for the truck to reach the same position as the car after it starts accelerating. We can use the equation of motion:
s = ut + (1/2)at^2
where:
s = displacement
u = initial velocity
a = acceleration
t = time
For the car, its initial velocity is 0 m/s because it was stationary, and the acceleration is 2.6 m/s^2. Let's find the time it takes for the car to reach the same position as the truck:
s = ut + (1/2)at^2
1.2x10^2 m = 0 + (1/2)(2.6 m/s^2)t^2
1.2x10^2 m = 1.3 m/s^2 * t^2
Simplifying the equation, we have:
t^2 = (1.2x10^2 m) / (1.3 m/s^2)
t^2 ≈ 92.31 s^2
Taking the square root of both sides, we get:
t ≈ 9.61 s
Therefore, it will take approximately 9.61 seconds for the truck to pass the car.
b) How far will the car have traveled when the truck passes it?
To find this, we can use the equation of motion again. This time, we'll calculate the displacement of the car using the time we just found (9.61 s):
s = ut + (1/2)at^2
s = (1/2)(2.6 m/s^2)(9.61 s)^2
Simplifying the equation, we have:
s ≈ 116 m
Therefore, the car will have traveled approximately 116 meters when the truck passes it.
c) If the car were to maintain this acceleration, how fast would it be traveling when it overtakes the truck?
In this case, we need to determine the time it takes for the car to catch up to the truck. The truck is traveling at a constant speed of 28.0 m/s, so the distance it needs to cover is the same as the distance the car has already traveled (116 meters).
Using the equation of motion once again, we can find the time it takes for the car to catch up:
s = ut + (1/2)at^2
116 m = 0 + (1/2)(2.6 m/s^2)t^2
116 m = 1.3 m/s^2 * t^2
Simplifying the equation, we have:
t^2 = (116 m) / (1.3 m/s^2)
t^2 ≈ 89.23 s^2
Taking the square root of both sides, we get:
t ≈ 9.45 s
Therefore, if the car maintains this acceleration, it will take approximately 9.45 seconds to overtake the truck. However, to calculate the car's final speed, we need to know its initial speed. This information is missing from the given data, so we cannot determine the car's final velocity.