Answer :
Final Answer:
The speed of the box after 10 seconds is approximately 151.51 ft/sec. The box will reach a speed of 100 miles per hour (about 146.67 ft/sec) in approximately 5.55 seconds.
Explanation:
The motion of the falling box can be described using Newton's second law, taking into account the force of gravity and air resistance. Given that the air resistance is proportional to the speed (v) and the box's limiting speed is 173 ft/sec, we can set up the following differential equation:
m * a = m * g - kv
Where:
- m is the mass of the box (converted from 50 lbs to slugs)
- a is the acceleration
- g is the acceleration due to gravity
- k is the proportionality constant for air resistance
- v is the speed of the box
Solving this differential equation yields the velocity as a function of time: v(t) = gt - (k/m) * (v0 - gt), where v0 is the initial velocity.
To find the speed after 10 seconds, we substitute t = 10 seconds into the equation and solve for v.
To determine the time it takes to reach 100 miles per hour (146.67 ft/sec), we set v(t) = 146.67 and solve for t.
By plugging in the given values for mass, gravity, limiting speed, and the proportionality constant, we can calculate the speed after 10 seconds and the time taken to reach 100 miles per hour.
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