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Chapter 4: Problem 43

An elevator ascends with an upward acceleration of \(4.0-\mathrm{ft} / \mathrm{s}^{2}\). At the instant its upward speed is \(8.0 \mathrm{ft} / \mathrm{s}\), a loose bolt drops from the ceiling of the elevator \(9.0 \mathrm{ft}\) from the floor. Calculate (a) the time of flight of the bolt from ceiling to floor and \((b)\) the distance it has fallen relative to the elevator shaft.

Short Answer

Expert verified
The time of flight of the bolt from the ceiling to the floor is the smaller of the two roots obtained from step 3, and the distance the bolt has fallen relative to the elevator shaft is obtained in step 4.

Step by step solution

01

Determine the initial speed of the bolt

The bolt has the same initial upward speed as the elevator when it is dropped, which is \(8.0 \, ft/s\).
02

Calculate the time of flight of the bolt

The bolt is under the effect of two accelerations: the acceleration due to gravity, which is \(32.2 \, ft/s^2\) in the downward direction, and the elevator's acceleration of \(4.0 \, ft/s^2\) in the upward direction. Consequently, the net acceleration is \(32.2 - 4.0 = 28.2 \, ft/s^2\) in the downward direction. Use the formula: \[ h = v_0t + 0.5at^2 \] where \(h\) is the height, \(v_0\) is the initial velocity, \(a\) is the acceleration, and \(t\) is the time to solve for \(t\): \[ 9.0 = 8.0t + 0.5(-28.2)t^2 \] Solve the above equation for \(t\) to find both roots.
03

Select the correct time root

Select the smaller positive root of \(t\) as the falling object cannot take negative time or a longer time which is the larger root to fall.
04

Calculate the relative distance from elevator shaft

The distance the bolt falls relative to the elevator shaft can be calculated by considering the motion of the bolt from the point of view of an observer in the elevator. From the frame of reference of the elevator, the bolt has an initial velocity of \(8.0 \, ft/s\) and a constant acceleration of \(32.2 \, ft/s^2\) downward (the acceleration due to gravity). So the net distance fallen relative can be calculated using \(h = v_0t + 0.5at^2\) where \(h\) is the distance, \(v_0\) is the initial velocity which is \(8.0 \, ft/s\), \(a\) is the acceleration, which is \(32.2 \, ft/s^2\), and \(t\) is the time obtained in step 3. Calculate \(h\) by plugging in these values.

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