7) You throw a 12-N stone vertically into the air from ground level. You observe that when it is 15.0 m above the ground, it is traveling at 25.0 m/s upward. Use the work –energy theorem to find (a) the stone’s speed just as it left the ground and (b) its maximum height.
8) A 6.0-kg block initially at rest is pulled to the right along a frictionless, horizontal surface by a constant horizontal force of 12 N. Use the work-kinetic energy theorem to find the block’s speed after it has moved 3.0 m.
7a. V^2 = Vo^2 + 2g*h
Vo^2 = V^2 - 2g*h = 25^2 + 19.6*15 = 919
Vo = 30.32 m/s.
7b. h = -(Vo^2)/2g = -(30.32^2)/-19.6 =
46.9 m.
8. a = F/m = 12/6 = 2 m/s^2
V^2 = Vo^2 + 2a*d
Vo = 0,
a = 2 m/s^2
d = 3 m.
Solve for V.
To solve these problems, we can use the work-energy theorem, which states that the net work done on an object is equal to the change in its kinetic energy.
(a) To find the stone's speed just as it left the ground, we need to determine its initial kinetic energy. We know that the stone is initially at rest, so its initial kinetic energy is zero. The net work done on the stone is equal to the change in kinetic energy.
The net work done on the stone is given by the formula:
Net work = Final kinetic energy - Initial kinetic energy
In this case, the final kinetic energy is given by:
Final kinetic energy = 1/2 * m * v^2
where m is the mass of the stone and v is its final velocity.
However, we don't have enough information to directly calculate the final kinetic energy or the stone's mass. But we can use the work-energy theorem to find the velocity just as it left the ground.
The work done on the stone is equal to the change in its kinetic energy:
Work = Final kinetic energy - Initial kinetic energy
Since the stone is thrown vertically into the air, the work done on it is zero (as no external force is acting). Therefore, the initial kinetic energy is equal to the final kinetic energy.
0 = 1/2 * m * v^2
Solving for v, we find that the speed of the stone just as it left the ground is 0 m/s (or simply, it's at rest).
(b) To find the maximum height reached by the stone, we can use the conservation of mechanical energy. At the highest point of its trajectory, the stone has maximum potential energy (due to its height) and no kinetic energy.
The work-energy theorem can be expressed as:
Net work = change in kinetic energy + change in potential energy
Since the stone is at rest at its maximum height, its change in kinetic energy is zero.
0 = change in kinetic energy + change in potential energy
We are given the final velocity at a point 15.0 m above the ground, which is 25.0 m/s. The initial velocity is zero.
Therefore, the change in potential energy is equal to the work done on the stone by gravity:
Change in potential energy = m * g * h
where m is the mass of the stone, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the maximum height.
Plugging in the values:
0 = 0 + m * 9.8 m/s^2 * h
Simplifying, we find:
h = 0 m
Based on the equation, the maximum height of the stone is zero, meaning it doesn't reach any height and falls back to the ground.
Moving on to the second problem:
To find the block's speed after it has moved 3.0 m, we can also use the work-energy theorem.
The work done on the block is equal to the change in its kinetic energy:
Work = Final kinetic energy - Initial kinetic energy
Since the surface is frictionless, there is no additional work done against friction. Therefore, the work done is equal to the change in kinetic energy.
Work = Change in kinetic energy
The work done on an object is given by the formula:
Work = Force * Distance * cos(theta)
In this case, the force applied is 12 N, and the distance moved is 3.0 m. Theta in this case would be 0 degrees since the force is acting horizontally.
We can plug in the values into the equation:
Work = 12 N * 3.0 m * cos(0 degrees)
The cosine of 0 degrees is 1, so the formula simplifies to:
Work = 12 N * 3.0 m * 1 = 36 J (Joules)
Now, we need to find the change in kinetic energy.
Change in kinetic energy = Work
The change in kinetic energy is equal to the final kinetic energy minus the initial kinetic energy.
Assuming the initial velocity is zero, and the final velocity is the speed we are trying to find, we have:
Change in kinetic energy = 1/2 * m * v^2 - 1/2 * m * 0^2 = 1/2 * m * v^2
Therefore,
1/2 * m * v^2 = 36 J
We don't have the mass of the block, so we can't calculate the exact speed. But we can say that the work done on the block will result in an increase in its kinetic energy.