A car is traveling at a constant speed of 27.1 m/s on a highway. At the instant this car passes an entrance ramp, a second car enters the highway from the ramp. The second car starts from rest and has a constant acceleration. What acceleration must it maintain, so that the two cars meet for the first time at the next exit, which is 2.81 km away?
To travel 2.81 km, and reach the next exit at the same time, the time spent getting there must be
t = 2810/27.1 = 103.7 s.
Require that
(1/2) a t^2 = 27.1 t, and
solve for a, using the t from above.
a = 54.2/t = 0.523 m/s^2
Well, to find the acceleration needed for the second car, we can use a physics formula called the kinematic equation. But first, let me remind you that it's not a race, so there's no need to rush! Safety first!
Now, using the equation of motion, we have:
s = ut + (1/2)at^2,
where s is the distance traveled, u is the initial velocity, a is the acceleration, and t is the time.
Let's consider the time taken by the second car to reach the next exit:
s = ut + (1/2)at^2,
2.81 km = 0 + (1/2)at^2,
Converting kilometers to meters,
2.81 km = 2810 m.
Simplifying the equation, we get:
2810 = (1/2)at^2.
Now, we need to consider the motion of the first car. Since it's traveling at a constant speed, there's no acceleration involved. Therefore, the equation for the first car would be:
s = ut + 0,
2810 = 27.1t.
Rearranging the equation gives us:
t = 2810 / 27.1.
Now, let's substitute this value into the equation for the second car:
2810 = (1/2)a(2810 / 27.1)^2.
Calculating that expression would give us the acceleration needed for the second car to meet the first car at the exit.
But hey, remember, it's just a hypothetical scenario, and the most important thing is driving safely and responsibly. No need to be in a hurry!
To calculate the acceleration needed for the second car to meet the first car at the next exit, we can use the following steps:
Step 1: Convert the distance from kilometers to meters.
Given: Distance = 2.81 km
1 km = 1000 m
Distance = 2.81 km × 1000 m/km
Distance = 2810 m
Step 2: Set up the equation for the first car.
The first car is traveling at a constant speed of 27.1 m/s. The time it takes for the two cars to meet is the same for both cars, so we can use the equation: Distance = Speed × Time.
Distance = 27.1 m/s × Time₁
Step 3: Set up the equation for the second car.
The second car starts from rest, so its initial speed is 0 m/s. The acceleration is constant, so we can use the equation: Distance = Initial Velocity × Time + (1/2) × Acceleration × Time².
Distance = 0 × Time₂ + (1/2) × Acceleration × Time₂²
Distance = (1/2) × Acceleration × Time₂²
Step 4: Set up the equation for the meeting time.
Since the two cars meet at the same time, we can equate the equations from steps 2 and 3:
27.1 m/s × Time₁ = (1/2) × Acceleration × Time₂²
Step 5: Solve for the meeting time.
Time₁ = Time₂ (since they meet at the same time)
27.1 m/s = (1/2) × Acceleration × Time₂
Time₂ = 27.1 m/s × 2 / Acceleration
Step 6: Substitute the value of Time₂ into the equation from step 3.
Distance = (1/2) × Acceleration × (27.1 m/s × 2 / Acceleration)²
2810 m = (1/2) × Acceleration × (54.2 m/s)² / Acceleration²
2810 m = 0.5 × 54.2 m/s × 54.2 m/s / Acceleration
Step 7: Solve for the acceleration.
To find the acceleration, we can solve the above equation for Acceleration:
Acceleration = 0.5 × 54.2 m/s × 54.2 m/s / 2810 m
Acceleration ≈ 1.03 m/s²
Therefore, the second car must maintain an acceleration of approximately 1.03 m/s² in order to meet the first car at the next exit.
To find the acceleration of the second car, we can use the equations of motion. Let's break down the problem and solve it step by step:
Step 1: Convert the distance to meters
The distance between the entrance ramp and the next exit is given as 2.81 km. To solve the problem, we need to convert this distance to meters since the other measurements are also in meters. 1 km is equal to 1000 meters, so 2.81 km is equal to 2.81 * 1000 = 2810 meters.
Step 2: Find the time taken by the first car to reach the next exit
We can use the equation of motion: distance = initial velocity * time + (1/2) * acceleration * time^2
Here, the initial velocity is 27.1 m/s, the distance is 2810 meters, and we need to find the time. We can rearrange the equation as follows:
2810 = 27.1 * t + (1/2) * a * t^2
Step 3: Find the speed of the first car when it reaches the next exit
The speed of the first car will be the velocity with which it is traveling, which is constant at 27.1 m/s.
Step 4: Use relative velocity to find the acceleration of the second car
The relative velocity between the two cars when they meet will be zero since they meet at the same point. We can use the equation of relative velocity:
relative velocity = velocity of the second car - velocity of the first car
Here, the relative velocity is zero, the velocity of the first car is 27.1 m/s, and we need to find the velocity of the second car. Rearranging the equation:
0 = v2 - 27.1
v2 = 27.1
Step 5: Use the equations of motion to find the acceleration of the second car
We can use the equation of motion: velocity = initial velocity + acceleration * time
Here, we know the initial velocity of the second car is 0 m/s (initially at rest), the final velocity is 27.1 m/s, and we need to find the acceleration. Rearranging the equation:
27.1 = 0 + a * t
Since we found the time in step 2, we substitute it into the equation:
27.1 = a * t
Finally, we can solve for the acceleration by dividing both sides of the equation by t:
acceleration = 27.1 / t
By following these steps, you can find the acceleration that the second car must maintain so that the two cars meet for the first time at the next exit.