A race driver has made a pit stop to refuel. After refueling, he starts from rest and leaves the pit area with an acceleration whose magnitude is 6.3 m/s2; after 4.1 s he enters the main speedway. At the same instant, another car on the speedway and traveling at a constant velocity of 70.7 m/s overtakes and passes the entering car. The entering car maintains its acceleration. How much time is required for the entering car to catch up with the other car?

V entry = a t = 6.3*4.1 = 25.83 m/s

v = 25.83 + 6.3 t
x = 25.83 t + (1/2)(6.3) t^2
for the second car x = 70.7 t
so
70.7 t = 25.83 t + 3.15 t^2

3.15 t^2 - 44.87 t = 0

t (3.15 t -44.87) = 0
so they are at the same x at sthe start of course and at
t = 44.87/3.15

To find the time required for the entering car to catch up with the other car, we need to determine the distance traveled by each car during the given time interval.

First, let's calculate the distance traveled by the car on the speedway:

Distance = Velocity * Time
Distance = 70.7 m/s * 4.1 s

Calculating this:
Distance = 289.87 meters (rounded to two decimal places)

Now, let's determine the distance traveled by the entering car during the same time interval. Since the entering car starts from rest and undergoes constant acceleration, we can use the kinematic equation:

Distance = Initial Velocity * Time + (1/2) * Acceleration * Time^2

Since the entering car starts from rest, the initial velocity is 0 m/s. Plugging in the values:

Distance = (1/2) * 6.3 m/s^2 * (4.1 s)^2

Calculating this:
Distance = 53.25 meters (rounded to two decimal places)

Now, we need to find out how much further the entering car needs to travel to catch up with the other car. This can be calculated by subtracting the distance traveled by the entering car from the distance traveled by the car on the speedway:

Catch-up Distance = (289.87 m) - (53.25 m)

Calculating this:
Catch-up Distance = 236.62 meters (rounded to two decimal places)

Finally, we can determine the time required for the entering car to catch up with the other car using the equation:

Catch-up Distance = Velocity * Time

Plugging in the values:

236.62 m = (70.7 m/s) * Time

Solving for Time:

Time = 236.62 m / 70.7 m/s

Calculating this:
Time = 3.34 seconds (rounded to two decimal places)

Therefore, the entering car would take approximately 3.34 seconds to catch up with the other car.