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.
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.
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.