How Is the Acceleration of a Rocket Calculated When a Bolt Falls Off?

[Ritzycat]

Homework Statement

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 6.0s later.

Homework Equations

$v_x = v_0 + at$
$x_f = x_0 + v_0t + 1/2at^2$
$v_x^2 = v_0^2 + 2a(Δx)$

The Attempt at a Solution

The biggest part I'm confused about is whether or not the bolt had an initial positive velocity (I am setting down as negative). It says the bolt "falls off the side", implying it just falls down.

Position of bolt before it falls
$0m = x + 1/2(-9.8m/s^2)(6.0s)^2$
$x_i = 176.4m$

Acceleration of rocket
$176.4m = 1/2(a)(4.0s)^2$
$a = 22.05 m/s^2$

My answer is incorrect.

 

[Doc Al]

The biggest part I'm confused about is whether or not the bolt had an initial positive velocity (I am setting down as negative). It says the bolt "falls off the side", implying it just falls down.

The bolt will have the same velocity as the rocket at the moment it falls free. "Falls off the side" just means that it's not shot out or launched, but just let go.

 

[PeroK]

or not the bolt had an initial positive velocity (I am setting down as negative). It says the bolt "falls off the side", implying it just falls down.

If you were cycling along and you fell off your bike, would you have forward velocity?

 

[Ritzycat]

Fair enough. I got to go now, but I'll attempt the problem later tonight. Conceptually I understand that - but the wording sort of got to me. Thanks for clarifying!

 

[HalfLostAndSearching]

I would approach this problem by first clarifying any assumptions or uncertainties in the given information. In this case, it is unclear whether the bolt had an initial velocity or not. To account for this, I would propose two possible scenarios:

1. The bolt had an initial velocity of zero, meaning it was simply released from the rocket and fell straight down. In this case, the acceleration of the rocket can be calculated using the position of the bolt before it fell, as you have done in your attempt at a solution. However, the acceleration would be negative, as the rocket is moving upwards and the bolt is falling downwards. So the equation would be 0m = x + 1/2(-a)(6.0s)^2, giving an acceleration of -9.8 m/s^2.

2. The bolt had an initial velocity that was the same as the rocket's, meaning it was attached to the rocket and moved upwards for 4 seconds before falling off. In this case, the acceleration of the rocket can be calculated using the position of the bolt when it falls, as it would be at the same height as the rocket at that point. The equation would be 0m = x + 1/2(a)(6.0s)^2, giving an acceleration of 9.8 m/s^2.

It is also important to note that the equations provided in the homework statement are for motion in one dimension (in this case, vertical motion). If we want to consider the motion of the rocket in two dimensions (taking into account its horizontal velocity), we would need to use vector equations and consider the effects of gravity on both the vertical and horizontal components of its motion.

Overall, as a scientist, I would use the given information to make reasonable assumptions and use the appropriate equations to calculate the acceleration of the rocket in each scenario. From there, I would compare my results and determine which scenario is more likely based on the given information.