You drop a 1.90 kg book to a friend who stands on the ground at distance D = 11.8 m below with outstretched hands at distance d = 1.42 m above the ground.
a)What is the speed of the book when it reaches the hands?
b)If we substituted a second book with twice the mass, what would its speed be?
Assuming we aren't factoring in wind drag or anything like that, the way to solve this is simple. First, find the distance the book falls, which in this case is 11.8-1.42 m, which equals a distance fallen of 10.38 m. Then, we can determine the time it takes for the book to reach your friend by dividing twice the distance fallen (10.38 m) by the force of gravity on Earth (9.8 m/s) and then find the square root, which gives you ~1.455 seconds. Finally, you multiply the force of gravity by the time taken to determine the velocity, which comes out to ~14.26 m/s. And, to answer the second half of the problem, the answer will also be ~14.26 m/s, because mass does not affect acceleration.
To calculate the speed of the book when it reaches your friend's hands, we can use the principle of energy conservation. The potential energy of the book at its initial height will be converted into kinetic energy as it falls.
a) Speed of the book:
First, we need to find the initial potential energy of the book. The potential energy of an object near the Earth's surface can be expressed as follows:
Potential Energy (PE) = mass (m) * acceleration due to gravity (g) * height (h)
Given:
mass of the book (m) = 1.90 kg
height (h) = 11.8 m (book's initial height)
acceleration due to gravity (g) ≈ 9.8 m/s²
So, the potential energy (PE) of the book is:
PE = 1.90 kg * 9.8 m/s² * 11.8 m
Next, we'll use the principle of energy conservation to convert the potential energy into kinetic energy. At the bottom, all the potential energy will be converted to kinetic energy:
Kinetic Energy (KE) = Potential Energy (PE)
Since the kinetic energy of a moving object can be expressed as:
Kinetic Energy (KE) = 0.5 * mass (m) * velocity² (v)
We can equate the potential energy (PE) and kinetic energy (KE) to solve for the velocity (v):
0.5 * mass (m) * velocity² (v) = Potential Energy (PE)
Substituting the values:
0.5 * 1.90 kg * velocity² (v) = 1.90 kg * 9.8 m/s² * 11.8 m
Now, solve for the velocity (v) by simplifying the equation:
velocity² (v) = (1.90 kg * 9.8 m/s² * 11.8 m) / (0.5 * 1.90 kg)
Finally, take the square root of both sides of the equation to find the velocity (v):
v = √[(1.90 kg * 9.8 m/s² * 11.8 m) / (0.5 * 1.90 kg)]
By evaluating the expression, we can determine the speed (v) of the book when it reaches your friend's hands.
b) If we substituted a second book with twice the mass:
If we substitute a second book with twice the mass, the new mass (m) would be 2 * 1.90 kg = 3.80 kg. To find the speed (v) of the heavier book using the same method, we can repeat the calculations with the new mass value (3.80 kg).
Simply substitute the new mass value and solve for the velocity (v) using the equation:
v = √[(new mass * 9.8 m/s² * 11.8 m) / (0.5 * new mass)]
Evaluate the expression to find the new speed (v) of the heavier book.
To find the speed of the book when it reaches the hands, we can use the principles of conservation of energy. The potential energy the book has at the initial height will be converted into kinetic energy as it falls.
We can use the equation:
Potential Energy (PE) = Kinetic Energy (KE)
a) Finding the speed of the 1.90 kg book:
Step 1: Calculate the potential energy at the initial height:
PE = m * g * h
= 1.90 kg * 9.8 m/s² * (11.8 m + 1.42 m)
= 1.90 kg * 9.8 m/s² * 13.22 m
≈ 244.68 J
Step 2: Convert all the potential energy into kinetic energy:
KE = PE
KE = 244.68 J
Step 3: Calculate the kinetic energy using the equation:
KE = 1/2 * m * v²
244.68 J = 1/2 * 1.9 kg * v²
v² = (244.68 J) / (1/2 * 1.9 kg)
v² = (244.68 J) / (3.8 kg)
v² ≈ 64.3842 m²/s²
Step 4: Take the square root of both sides to find the velocity:
v ≈ √(64.3842 m²/s²)
v ≈ 8.03 m/s
Therefore, the speed of the book when it reaches the hands is approximately 8.03 m/s.
b) Now, let's calculate the speed of the book if we substitute a second book with twice the mass:
Step 1: Calculate the potential energy at the initial height:
PE = m * g * h
= (2 * 1.90 kg) * 9.8 m/s² * (11.8 m + 1.42 m)
= 3.8 kg * 9.8 m/s² * 13.22 m
≈ 489.36 J
Step 2: Convert all the potential energy into kinetic energy:
KE = PE
KE = 489.36 J
Step 3: Calculate the kinetic energy using the equation:
KE = 1/2 * m * v²
489.36 J = 1/2 * (2 * 1.9 kg) * v²
v² = (489.36 J) / (1.9 kg)
v² ≈ 257.56 m²/s²
Step 4: Take the square root of both sides to find the velocity:
v ≈ √(257.56 m²/s²)
v ≈ 16.04 m/s
Therefore, if we substitute a second book with twice the mass, its speed when it reaches the hands will be approximately 16.04 m/s.