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Chapter 4: Problem 479

A spherical ball of mass \(15 \mathrm{~kg}\) stationary at the top of a hill of height \(82 \mathrm{~m} .\) It slides down a smooth surface to the ground, then climbs up another hill of height \(32 \mathrm{~m}\) and finally slides down to horizontal base at a height of \(10 \mathrm{~m}\) above the ground. The velocity attained by the ball is (A) \(30 \sqrt{10 \mathrm{~m} / \mathrm{s}}\) (B) \(10 \sqrt{30 \mathrm{~m} / \mathrm{s}}\) (C) \(12 \sqrt{10} \mathrm{~m} / \mathrm{s}\) (D) \(10 \sqrt{12} \mathrm{~m} / \mathrm{s}\)

Short Answer

Expert verified
The velocity attained by the ball is approximately \(10\sqrt{30}\) m/s.

Step by step solution

01

Total Energy at the Initial Height

At first, we need to analyze the total energy of the ball at the initial height (82 meters). At this height, the gravitational potential energy is maximum, and the kinetic energy is zero as the ball is stationary. The potential energy PE can be calculated as follows: PE = m × g × h Here, m = 15 kg (mass of the ball) g = 9.81 m/s² (acceleration due to gravity) h = 82 m (initial height) PE = 15 kg × 9.81 m/s² × 82 m PE = 12015 J (Joules)
02

Energy Conservation between First Hill and Second Hill

As the ball rolls down the first hill and climbs the second hill, we can use the conservation of energy law to calculate the velocity of the ball after passing the second hill height (32 meters). We can assume the Potential Energy at the height of 10 meters is PE10 = m × g × 10 Moreover, the Potential Energy at the height of 32 meters is PE32 = m × g × 32 The Kinetic energy KE10 and KE32 at heights 10 meters and 32 meters can be calculated using the following equation: KE10 = PE - PE10 KE32 = PE - PE32 Since the velocity is the same at 10 meters and 32 meters height, the Kinetic energies must be equal: KE10 = KE32 Therefore, the Potential Energies at these heights should follow: PE10 - PE = PE32 - PE
03

Calculate the Potential Energy at Heights

Using the equations from Step 2, we can calculate the Potential Energy at different heights: PE10 = m × g × 10 PE10 = 15 kg × 9.81 m/s² × 10 m PE10 = 1471.5 J PE32 = m × g × 32 PE32 = 15 kg × 9.81 m/s² × 32 m PE32 = 4716 J
04

Calculate the Kinetic Energy and the Final Velocity

After calculating the Potential Energies, we can calculate the Kinetic Energy and then the final velocity: KE10 = PE - PE10 KE10 = 12015 J - 1471.5 J KE10 = 10543.5 J Since KE10 = KE32, we have the Kinetic Energy at the height of 32 meters: KE32 = 10543.5 J Now, we calculate the velocity of the ball using the relation KE = (1/2)mv²: 10543.5 J = (1/2) × 15 kg × v² Dividing both sides by 7.5 kg, we get: v² = 1405.8 Taking the square root of both sides: v = \(\sqrt{1405.8}\) m/s v ≈ 37.5 m/s Comparing the values to the given options, the final velocity is closest to \(10\sqrt{30}\) m/s, which is option (B).

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