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Chapter 6: Problem 40

A classmate throws a \(1.0-\mathrm{kg}\) book from a height of \(1.0 \mathrm{~m}\) above the ground straight up into the air. The book reaches a maximum height of \(3.0 \mathrm{~m}\) above the ground and begins to fall back. Assume that \(1.0 \mathrm{~m}\) above the ground is the reference level for zero gravitational potential energy. Determine a) the gravitational potential energy of the book when it hits the ground. b) the velocity of the book just before hitting the ground.

Short Answer

Expert verified
Answer: The gravitational potential energy of the book when it hits the ground is 0 J, and its velocity just before hitting the ground is 6.26 m/s.

Step by step solution

01

Determine the change in gravitational potential energy

The book was initially thrown from a height of 1.0 m and reached a maximum height of 3.0 m before falling back to the ground. We are asked to find the gravitational potential energy of the book when it hits the ground. As we have taken 1.0 m above the ground as the reference level for zero gravitational potential energy, when the book hits the ground its height with respect to the reference level is 0. Therefore, the change in gravitational potential energy is given by PE = mgh = (1.0 kg) * (9.81 m/s²) * (0 m) = 0 J. So, the gravitational potential energy when it hits the ground is 0 J. a) Gravitational potential energy when it hits the ground: 0 J
02

Calculate the potential and kinetic energy at the highest point

We need to determine the potential and kinetic energy of the book at the highest point to find the total mechanical energy. The potential energy at the highest point is given by PE = mgh = (1.0 kg) * (9.81 m/s²) * (2.0 m) = 19.62 J (As the maximum height is 3.0 m, the difference in height from the reference level is 3.0 m - 1.0 m = 2.0 m). At the highest point, the book momentarily comes to a stop, so its kinetic energy is 0. Potential energy at the highest point: 19.62 J Kinetic energy at the highest point: 0 J
03

The Conservation of Mechanical Energy

When the book is just before hitting the ground, the potential energy is 0. Then, by conservation of mechanical energy, the kinetic energy becomes equal to the mechanical energy we found at the highest point. Hence, KE = 19.62 J. At the highest point: KE_high + PE_high = KE_ground + PE_ground
04

Determine the book's velocity just before hitting the ground

Now we can determine the book's velocity just before hitting the ground. We know the kinetic energy at this point is 19.62 J. Using the formula KE = (1/2)mv², we can solve for v. 19.62 J = (1/2)(1.0 kg)v² Rearrange the formula to get v²: v² = (2 * 19.62 J) / (1.0 kg) = 39.24 Now, take the square root to find the velocity: v = sqrt(39.24) = 6.26 m/s b) The book's velocity just before hitting the ground: 6.26 m/s

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