11. A 3 stage rocket is launched vertically upwards. The data for each stage is as follows:
STAGE ACCELERATION /TIME
1 15m/s 2 60s
2 20m/s 2 40s
3 l0m/s 2 80s
Determine: (a) the maximum velocity of the climbing rocket. (b) the height of the rocket at the
end of the third stage. ©) the maximum height attained by the rocket.
To solve this problem, we need to use the equations of motion and apply them to each stage of the rocket.
(a) To find the maximum velocity of the rocket, we need to find the final velocity of the third stage. We can use the equation:
vf = vi + at
For stage 1:
vi = 0 (as the rocket starts from rest)
a1 = 15 m/s^2
t1 = 60 s
Using the equation, we get:
v1 = 0 + (15 m/s^2)(60 s) = 900 m/s
For stage 2:
vi = v1 (the final velocity from stage 1)
a2 = 20 m/s^2
t2 = 40 s
Using the equation, we get:
v2 = 900 m/s + (20 m/s^2)(40 s) = 1700 m/s
For stage 3:
vi = v2 (the final velocity from stage 2)
a3 = 10 m/s^2
t3 = 80 s
Using the equation, we get:
v3 = 1700 m/s + (10 m/s^2)(80 s) = 2500 m/s
Therefore, the maximum velocity of the climbing rocket is 2500 m/s.
(b) To find the height of the rocket at the end of the third stage, we can use the equation:
s = vit + (1/2)at^2
For stage 1:
vi = 0 (as the rocket starts from rest)
a1 = 15 m/s^2
t1 = 60 s
Using the equation, we get:
s1 = (0)(60 s) + (1/2)(15 m/s^2)(60 s)^2 = 27000 m
For stage 2:
vi = v1 (the final velocity from stage 1)
a2 = 20 m/s^2
t2 = 40 s
Using the equation, we get:
s2 = 27000 m + (900 m/s)(40 s) + (1/2)(20 m/s^2)(40 s)^2 = 63000 m
For stage 3:
vi = v2 (the final velocity from stage 2)
a3 = 10 m/s^2
t3 = 80 s
Using the equation, we get:
s3 = 63000 m + (1700 m/s)(80 s) + (1/2)(10 m/s^2)(80 s)^2 = 206000 m
Therefore, the height of the rocket at the end of the third stage is 206000 m.
(c) To find the maximum height attained by the rocket, we need to find the maximum height reached during each stage. The maximum height during each stage can be calculated using the equation:
hmax = (vf^2 - vi^2) / (2a)
For stage 1:
vi = 0 (as the rocket starts from rest)
vf = v1 (the final velocity from stage 1)
a1 = 15 m/s^2
Using the equation, we get:
hmax1 = (900 m/s)^2 / (2 * 15 m/s^2) = 27000 m
For stage 2:
vi = v1 (the final velocity from stage 1)
vf = v2 (the final velocity from stage 2)
a2 = 20 m/s^2
Using the equation, we get:
hmax2 = (1700 m/s)^2 / (2 * 20 m/s^2) = 72250 m
For stage 3:
vi = v2 (the final velocity from stage 2)
vf = v3 (the final velocity from stage 3)
a3 = 10 m/s^2
Using the equation, we get:
hmax3 = (2500 m/s)^2 / (2 * 10 m/s^2) = 312500 m
Therefore, the maximum height attained by the rocket is 312500 m.