Two cars both travel along a straight road Northbound. At t = 0, the car 1 is 190.6 m further north than car 2. Also at t = 0, car 1 is travelling at 24.3 m/s, and car 2 is travelling at 5.7 m/s. Car 1 is slowing down with an acceleration of magnitude 0.28 while car 2 is speeding up with an acceleration of magnitude 0.25.
When car 2 reaches car 1, how fast is car 2 moving? (Answer in m/s to three sig digs.)
To find out how fast car 2 is moving when it reaches car 1, we need to calculate the time it takes for car 2 to catch up to car 1. Once we have the time, we can determine the final velocity of car 2. Let's break down the problem step by step.
Step 1: Determine the relationship between the positions of the two cars at any given time, t.
Since both cars are traveling along the same straight road, the position of each car can be described by the equation:
Position of car 1 = Initial position of car 1 + (velocity of car 1 * time) - (0.5 * acceleration of car 1 * time^2)
Position of car 2 = Initial position of car 2 + (velocity of car 2 * time) + (0.5 * acceleration of car 2 * time^2)
Step 2: Set up the problem using the given data.
For car 1:
Initial position of car 1 = 0 m (since it is the reference point)
Velocity of car 1 = 24.3 m/s (given)
Acceleration of car 1 = -0.28 m/s^2 (negative value indicates deceleration)
For car 2:
Initial position of car 2 = -190.6 m (since it starts behind car 1)
Velocity of car 2 = 5.7 m/s (given)
Acceleration of car 2 = 0.25 m/s^2
Step 3: Find the time when the two cars meet.
We set up the equation by equating the positions of the two cars:
0 + (24.3 * t) - (0.5 * 0.28 * t^2) = -190.6 + (5.7 * t) + (0.5 * 0.25 * t^2)
Simplifying the equation, we get:
0.14t^2 - 18.6t + 190.6 = 0
We can solve this quadratic equation to find the time, t. Using the quadratic formula, we get:
t = (-(-18.6) ± √((-18.6)^2 - 4 * 0.14 * 190.6)) / 2 * 0.14
Solving this equation gives us two possible solutions for t: t1 = 82.17 s and t2 ≈ -1.14 s. Since negative time is not meaningful in this context, we discard the negative value.
Therefore, it will take approximately 82.17 seconds for car 2 to catch up to car 1.
Step 4: Calculate the final velocity of car 2.
To find the final velocity of car 2, we use the equation:
Final velocity of car 2 = Velocity of car 2 + (acceleration of car 2 * time)
Substituting the given values, we have:
Final velocity of car 2 = 5.7 + (0.25 * 82.17)
Calculating this, we get:
Final velocity of car 2 ≈ 26.39 m/s
So, when car 2 reaches car 1, it will be moving at approximately 26.39 m/s (to three significant figures).