The tomato is dropped. What is the velocity, v , of the tomato when it hits the ground? Assume 91.9% of the work done in Part A is transferred to kinetic energy, E , by the time the tomato hits the ground.
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To find the velocity of the tomato when it hits the ground, we can use the principle of conservation of energy.
The potential energy of the tomato at the starting position is converted into kinetic energy just before it hits the ground.
Let's assume the initial potential energy of the tomato is Ep and the final kinetic energy is Ek.
According to the problem, 91.9% of the work done in Part A is transferred to kinetic energy. This means the final kinetic energy (Ek) is equal to 91.9% of the initial potential energy (Ep).
Mathematically, we can write this as:
Ek = 0.919 * Ep
The potential energy of an object near the surface of the Earth can be calculated using the equation:
Ep = m * g * h
Where:
m = mass of the tomato
g = acceleration due to gravity (approximately 9.8 m/s^2)
h = height from which the tomato is dropped
Now, let's say the mass of the tomato is m and the height from which it is dropped is h.
Substituting these values in the equation for potential energy, we get:
Ep = m * g * h
And substituting this expression for Ep in the equation for kinetic energy, we have:
Ek = 0.919 * (m * g * h)
Now, the kinetic energy of an object is given by the equation:
Ek = (1/2) * m * v^2
Where:
v = velocity of the tomato when it hits the ground
Plugging in the expression for Ek in the equation for kinetic energy, we get:
(1/2) * m * v^2 = 0.919 * (m * g * h)
Now, let's solve this equation for v.
To determine the velocity of the tomato when it hits the ground, we need to use the concept of conservation of mechanical energy.
First, let's define the variables:
- v = velocity of the tomato when it hits the ground (what we want to find)
- E = kinetic energy of the tomato when it hits the ground
- W = work done on the tomato during its fall
According to the problem, 91.9% of the work done on the tomato is transferred to kinetic energy, so we have:
E = 0.919 * W
Next, we need to establish the relationship between work done and kinetic energy. Work done on an object is equal to the change in its kinetic energy. In this case, the tomato starts from rest, so the initial kinetic energy is zero:
W = E
Since we know that E = 0.919 * W, we can substitute this into the equation:
W = 0.919 * W
Solve for W:
1 * W = 0.919 * W
W = 0.919 * W
Now, we can use the work-energy principle to find the velocity of the tomato when it hits the ground. The work done on an object is equal to the change in its kinetic energy:
W = ΔKE
We can rewrite this equation as:
W = KE_final - KE_initial
Since the tomato starts from rest, its initial kinetic energy is zero, so:
W = KE_final - 0
Therefore:
W = KE_final
We can substitute W with 0.919 * W:
0.919 * W = KE_final
Now, we have the equation for the final kinetic energy of the tomato.
To determine the velocity, we use the equation for kinetic energy:
KE = (1/2) * m * v^2
Where m is the mass of the tomato.
Rearranging the equation to solve for velocity, we get:
v = √(2 * KE / m)
Substituting the value of KE_final as 0.919 * W and rearranging the equation, we have:
v = √(2 * (0.919 * W) / m)
Now, we can solve for the velocity of the tomato when it hits the ground.