a hopper jumps straight up to a height of 0.45m. with what velocity will it return to the table?
.45 times 9.8 meters/seconds squared (which is free fall gravity rate)
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Well, I must say, that hopper really gets some "hopping" done! To determine the velocity at which it will return to the table, let's analyze the situation.
Assuming no external forces like air resistance, we can use the principle of conservation of energy. At the maximum height of 0.45m, all of the hopper's initial kinetic energy is converted into gravitational potential energy. When the hopper returns to the table, this potential energy will be completely converted back into kinetic energy.
So, we can set up an equation using the conservation of energy:
Initial Kinetic Energy = Final Kinetic Energy
But since the hopper is initially at rest on the table, its initial kinetic energy is zero. Therefore, the final kinetic energy when it returns to the table will also be zero.
This means that the velocity at which the hopper returns to the table will be zero. It will be momentarily at rest until the next "hopping" adventure begins!
To determine the velocity with which the hopper returns to the table, we need to understand the concept of conservation of energy. The hopper's starting and ending heights are the same, which means that its potential energy at the top is converted into kinetic energy when it returns to the table.
The potential energy (PE) of an object at a certain height is given by the formula:
PE = mgh
Where m represents the mass of the object, g is the acceleration due to gravity, and h is the height. Since the height is 0.45m, we can plug in the values:
PE = m * 9.8 * 0.45
Now, since the potential energy is converted into kinetic energy when the hopper returns to the table, we can equate the potential energy to kinetic energy:
PE = KE
KE = (1/2)mv^2
Where v represents the velocity of the hopper. By setting PE equal to KE, we can solve for v:
mgh = (1/2)mv^2
Canceling the mass (m) and multiplying both sides by 2, we get:
2gh = v^2
Taking the square root of both sides, we find:
v = √(2gh)
Now we can calculate the velocity by substituting the known values:
v = √(2 * 9.8 * 0.45)
Simplifying, we get:
v ≈ 3.14 m/s
Therefore, the hopper will return to the table with a velocity of approximately 3.14 m/s.