You drop a 1.8 kg textbook to a friend who stands on the ground 9 m below the textbook with outstretched hands 1.50 m above the ground.
(a) What is the speed of the textbook when it reaches the hands?
m/s
(b) If we substituted a second textbook with twice the mass, what would its speed be?
m/s
(c) If, instead, the textbook were thrown down, would the answer to (a) increase, decrease, or remain the same?
I will be happy to critique your thinking on this.
What did Galelo prove at Pisa?
Galileo Galilei famously demonstrated his law of falling bodies at the Leaning Tower of Pisa in 1589. He showed that all objects, regardless of their mass, fall at the same rate of acceleration due to gravity. This was a revolutionary idea at the time, as it contradicted the Aristotelian view that heavier objects fall faster than lighter objects.
To calculate the speed of the textbook when it reaches the hands of your friend and answer the other related questions, we can use basic kinematics equations.
(a) To find the speed of the textbook when it reaches the hands, we can use the equation for motion in the vertical direction:
v^2 = u^2 + 2as
Where:
v = final velocity (speed of the textbook)
u = initial velocity (0 m/s since the book was released from rest)
a = acceleration due to gravity (-9.8 m/s^2, as it acts downwards)
s = displacement (9 m)
Plugging in the values, the equation becomes:
v^2 = 0^2 + 2*(-9.8)*9
Simplifying:
v^2 = 0 + (-176.4)
v^2 = -176.4
Since speed cannot be negative, we take the positive square root, resulting in:
v ≈ 13.27 m/s
So, the speed of the textbook when it reaches the hands is approximately 13.27 m/s.
(b) If we substitute a second textbook with twice the mass (1.8 kg * 2 = 3.6 kg), the speed of the second textbook will remain the same. The mass of an object does not affect its acceleration due to gravity. Both objects will fall towards the ground with the same acceleration, resulting in the same speed when they reach the hands.
Therefore, the speed of the second textbook would also be approximately 13.27 m/s.
(c) If the textbook were thrown down instead of being released from rest, the answer to (a) would increase. This is because when the textbook is thrown, it gains an initial downward velocity. As a result, it would accumulate more speed as it falls towards the ground, leading to a higher final speed when it reaches the hands compared to simply being dropped.
(a) To find the speed of the textbook when it reaches the hands, we can use the equations of motion. The equation that relates the initial velocity (Vi), acceleration (a), displacement (Δy), and final velocity (Vf) is:
Vf² = Vi² + 2aΔy
In this case, the initial velocity (Vi) is zero because the textbook is dropped, and the acceleration (a) is the acceleration due to gravity (9.8 m/s²). The displacement (Δy) is the total distance traveled by the textbook from the initial position to the hands, which is the sum of the initial height (9 m) and the height of the outstretched hands (1.50 m).
Using the equation and plugging in the values, we can calculate the speed (Vf) of the textbook when it reaches the hands:
Vf² = 0² + 2(9.8 m/s²)(9 m + 1.50 m)
Vf² = 2(9.8 m/s²)(10.5 m)
Vf² = 205.8 m²/s²
Taking the square root of both sides, we get:
Vf = √(205.8 m²/s²)
Vf ≈ 14.34 m/s
Therefore, the speed of the textbook when it reaches the hands is approximately 14.34 m/s.
(b) If we substitute a second textbook with twice the mass, the speed (Vf) of the second textbook when it reaches the hands can be calculated using the same equation. The acceleration due to gravity remains the same, but the mass (m) of the second textbook is doubled.
Using the equation and plugging in the modified values:
Vf² = 0² + 2(9.8 m/s²)(10.5 m)
Vf² = 205.8 m²/s²
Therefore, the speed of the second textbook when it reaches the hands would also be approximately 14.34 m/s, as the mass of the object does not affect the speed when dropped from the same height.
(c) If the textbook were thrown down instead of being dropped, the answer to part (a) would remain the same. The speed of the textbook when it reaches the hands is determined solely by the height it falls from and the acceleration due to gravity. The initial velocity in this case would be non-zero, but it would not affect the final speed since the acceleration due to gravity remains the same.