A 2.0-kg purse is dropped from the top of the Leaning Tower of Pisa and falls 55 m before reaching the ground with a speed of 27 m/s. What was the average force of air resistance?
The correct answer and explanation is:
To find the average force of air resistance, we compare the actual energy change (including air resistance) to the ideal case (free fall without air resistance). Here’s how to approach it:
Step 1: Known Values
- Mass, m=2.0 kgm = 2.0 \, \text{kg}
- Height, h=55 mh = 55 \, \text{m}
- Final velocity, v=27 m/sv = 27 \, \text{m/s}
- Gravitational acceleration, g=9.8 m/s2g = 9.8 \, \text{m/s}^2
Step 2: Energy Without Air Resistance
If there were no air resistance, all the potential energy would convert into kinetic energy: Potential Energy=mgh=2.0×9.8×55=1078 J\text{Potential Energy} = mgh = 2.0 \times 9.8 \times 55 = 1078 \, \text{J} Kinetic Energy (actual)=12mv2=12×2.0×272=729 J\text{Kinetic Energy (actual)} = \frac{1}{2}mv^2 = \frac{1}{2} \times 2.0 \times 27^2 = 729 \, \text{J}
The difference in energy is due to air resistance: Energy lost to air resistance=1078−729=349 J\text{Energy lost to air resistance} = 1078 – 729 = 349 \, \text{J}
Step 3: Work Done by Air Resistance
Work done by air resistance = Force × Distance × cos(θ)
Here, air resistance is opposite to motion, so cos(180∘)=−1\cos(180^\circ) = -1: W=−Fair⋅d⇒Fair=−Wd=−−34955≈6.35 NW = -F_{\text{air}} \cdot d \Rightarrow F_{\text{air}} = -\frac{W}{d} = -\frac{-349}{55} \approx 6.35 \, \text{N}
So, the magnitude of the average force of air resistance is: 6.35 N\boxed{6.35 \, \text{N}}
300-word Explanation
To calculate the average force of air resistance acting on a 2.0-kg purse dropped from the Leaning Tower of Pisa, we use energy principles. In an ideal vacuum, gravity is the only force acting on a falling object, and thus all potential energy converts to kinetic energy. However, since the purse fell through air and reached a speed of 27 m/s rather than the higher speed expected in vacuum, energy was lost—specifically, it was dissipated due to air resistance.
First, we calculate the gravitational potential energy (PE) at the start: PE=mgh=2.0×9.8×55=1078 JPE = mgh = 2.0 \times 9.8 \times 55 = 1078 \, \text{J}. Then, we calculate the actual kinetic energy (KE) the purse had just before hitting the ground: KE=12mv2=12×2.0×272=729 JKE = \frac{1}{2}mv^2 = \frac{1}{2} \times 2.0 \times 27^2 = 729 \, \text{J}. The difference between these two energies—349 J—represents the energy lost to air resistance.
This energy loss is due to the work done by air resistance as it opposes the motion. Using the work-energy principle, we calculate the average force by dividing the energy lost by the distance fallen: F=34955≈6.35 NF = \frac{349}{55} \approx 6.35 \, \text{N}. This force acts opposite the direction of motion, so the purse ends with less speed than it would have had in a vacuum.
Therefore, the average force of air resistance is 6.35 newtons acting upward against the motion.