An ice sled powered by a rocket engine starts from rest on a large frozen lake and accelerates at +15.0 m/s2. At tl = 2.8 s the rocket engine is shut down and the sled moves with constant velocity v. At the 7800 m mark, the sled begins to accelerate at -7 m/s2. What is the final position of the sled when it comes to rest? What is the final position of the sled when it comes to rest?

V1 = a*t = 15 * 2.8 = 42 m/s.

d1 = 0.5a*t^2 = 0.5*15*2.8^2 = 58.8 m.

V^2 = V1^2 + 2a*d = 0.
d = -V1^2/2a = -(42^2)/-7 = 252 m.

Final position = 7800 + 252 = 8052 m.
mark.

To find the final position of the sled when it comes to rest, we need to break down the problem into different stages and calculate the distance traveled during each stage.

Stage 1: Acceleration from rest
First, we need to find the time it takes for the sled to reach the constant velocity v after the rocket engine is shut down. We can use the equation of motion:
v = u + at
where:
- v is the final velocity
- u is the initial velocity (0 m/s since the sled starts from rest)
- a is the acceleration (+15.0 m/s^2)
- t is the time

Rearranging the equation:
t = (v - u) / a
Substituting the given values:
t = (v - 0) / 15.0
t = v / 15.0

During the first stage, the distance traveled is given by the equation:
s1 = u * t + (1/2) * a * t^2
Substitute the known values:
s1 = 0 * t + (1/2) * 15.0 * t^2
s1 = (15/2) * t^2
s1 = (15/2) * (v/15.0)^2
s1 = (v^2) / 20

Stage 2: Constant velocity
During this stage, the sled moves with constant velocity v. The time traveled during this stage is tl - t since t is the time it took for the sled to reach v. Therefore, the distance traveled during this stage is:
s2 = v * (tl - t)

Stage 3: Retardation until rest
The sled begins to decelerate at -7 m/s^2 after it has traveled a distance of 7800 m. We need to find the time it takes for the sled to come to rest. Using the equation of motion:
v^2 = u^2 + 2as
where:
- v is the final velocity (0 m/s since the sled comes to rest)
- u is the initial velocity (v m/s since the sled is traveling with constant velocity v)
- a is the acceleration (-7 m/s^2)
- s is the distance (7800 m)

Rearranging the equation:
0 = v^2 + 2as
Simplifying:
v^2 = -2as
Substituting the known values:
0 = v^2 + 2(-7)(7800)
-15600 = v^2
v = √(15600)
v = 124.9 m/s

Now we can calculate the time it takes for the sled to come to rest using the equation:
t2 = (v - u) / a
Substituting the known values:
t2 = (0 - 124.9) / -7
t2 = 124.9 / 7
t2 = 17.8 s

The distance traveled during the third stage is given by the equation:
s3 = u * t2 + (1/2) * a * t2^2
Substituting the known values:
s3 = 124.9 * 17.8 + (1/2) * (-7) * (17.8^2)
s3 = 2220.22 m - 1395.83 m
s3 = 824.39 m

Final position of the sled:
The final position of the sled is the sum of the distances traveled during each stage:
Final position = s1 + s2 + s3
Final position = (v^2) / 20 + v * (tl - t) + 824.39

Note: The value of v is not given in the problem, so we cannot calculate the exact final position without that information.