A rocket is launched straight up with constant acceleration. Four seconds after liftoff, a bolt falls off the side of the rocket. The bolt hits the ground 6.10s later. What was the rocket's acceleration?
consider the bolt when on the rocket.
Vfbolt=a*4
hfbolt=1/2*a*16=8a
Now in the free fall portion, these become vi, and hi for the bolt
vfbolt=vibolt-g*6.1=4a-6.1g
hfbolt=6.1*4a-1/2 g 6.1^2+8a=0
from this second equation
you can solve for a.
To find the rocket's acceleration, we can use the equation of motion:
s = ut + (1/2)at^2
Where:
s = distance (height)
u = initial velocity
t = time
a = acceleration
Given:
t1 = 4 seconds (time when the bolt fell off)
t2 = 6.10 seconds (time for bolt to hit the ground)
s = unknown (distance)
Let's first find the distance traveled by the bolt from time t1 to t2.
Using the formula:
s = ut + (1/2)at^2
We know that the initial velocity is 0 (assuming no horizontal motion), so the equation becomes:
s = (1/2)at^2
Substituting the given values:
6.10 = (1/2)a(6.10)^2
Now, we can solve for 'a'.
6.10 = (1/2)a(6.10)^2
a = (2 * 6.10) / (6.10)^2
a ≈ 2 m/s^2
Therefore, the rocket's acceleration is approximately 2 m/s^2.
To find the rocket's acceleration, we can use the equations of motion. Let's break down the problem and work step-by-step to find the answer.
Step 1: Identify the known quantities:
- Time when the bolt hits the ground: t = 6.10 seconds
- Time elapsed since liftoff when the bolt falls off the rocket: t' = 4 seconds
- Initial velocity of the bolt: u = 0 m/s (since the bolt fell from rest)
Step 2: Determine the time it took for the bolt to reach the ground after falling off the rocket:
Since the bolt fell off the rocket after 4 seconds and hit the ground after 6.10 seconds, we can calculate the time taken by the bolt in the air.
t'' = t - t' = 6.10s - 4s = 2.10 seconds
Step 3: Calculate the displacement of the bolt:
Using the equation of motion for the displacement (s) with constant acceleration (a):
s = ut + (1/2)*a*t^2
Since the initial velocity of the bolt is zero, the equation simplifies to:
s = (1/2)*a*t^2
Substituting the values:
s = (1/2)*a*t''^2
Step 4: Calculate the displacement of the rocket during the same time:
The rocket has been accelerating for a total of 4 seconds (t'), so let's calculate its displacement during this time.
s' = (1/2)*a*t'^2
Step 5: Equate the two displacements:
Since both the bolt and the rocket traveled the same distance (or displacement) during the time the bolt was in the air (t''), we can equate s and s':
(1/2)*a*t''^2 = (1/2)*a*t'^2
Step 6: Solve for acceleration (a):
Divide both sides of the equation by (1/2) and cancel out:
a*t''^2 = a*t'^2
Dividing both sides by a:
t''^2 = t'^2
Taking the square root of both sides:
t'' = t'
Step 7: Substitute the values and solve for acceleration (a):
t'' = t' -> 2.10 seconds = 4 seconds
So, the rocket's acceleration is given by:
a = (2.10 seconds) / (4 seconds)
Calculating this division:
a = 0.525 m/s^2
Therefore, the rocket's acceleration is 0.525 m/s^2.