A construction worker is standing in a 2.6 m deep well, 3.1 m from the side of the well's wall. He tosses a hammer to a companion outside the well. If the hammer leaves his hand 1.0 m above the bottom of the well, at an angle of 35 degrees to the horizontal, what is the minimum speed it must have to clear the edge of the well? How far from the edge of the well does it land?
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To solve this problem, we can break it down into two parts: finding the minimum speed required for the hammer to clear the edge of the well and calculating the horizontal distance it will travel before landing outside the well.
Let's start by finding the minimum speed required for the hammer to clear the edge of the well:
1. Calculate the vertical distance the hammer needs to travel to clear the edge of the well:
- The total depth of the well is 2.6 m.
- The hammer leaves the worker's hand 1.0 m above the bottom of the well.
- Therefore, the vertical distance the hammer needs to travel is (2.6 - 1.0) = 1.6 m.
2. Calculate the horizontal distance the hammer needs to travel to reach the edge of the well:
- The worker is standing 3.1 m from the side of the well's wall.
- Therefore, the horizontal distance the hammer needs to travel is 3.1 m.
3. Calculate the angle at which the hammer is thrown (angle with respect to the horizontal):
- The given angle is 35 degrees.
4. Use the vertical and horizontal distances along with the angle to find the minimum speed:
- The minimum speed required can be calculated using the following kinematic equation:
v = sqrt((d * g) / sin(2θ)),
where v is the required minimum speed, d is the horizontal distance, g is the acceleration due to gravity (approximately 9.8 m/s^2), and θ is the angle in radians.
- Convert the angle from degrees to radians: θ = 35 * (π/180).
- Plug in the values into the formula and solve for v:
v = sqrt((3.1 * 9.8) / sin(2 * (35 * (π/180)))).
(Note that sin(2θ) = sin(70 * (π/180)) = sin(70°))
- Calculating this equation will give you the minimum speed required for the hammer to clear the edge of the well.
Now, let's calculate the horizontal distance the hammer will travel before landing outside the well:
1. Use the horizontal distance, initial vertical velocity (0 since the hammer is thrown horizontally), and the angle to calculate the time of flight using the kinematic equation:
- t = d / (v * cos(θ)),
where t is the time of flight, d is the horizontal distance, v is the minimum speed, and θ is the angle in radians.
- Plug in the values into the equation and solve for t.
2. Calculate the horizontal distance traveled by the hammer during the time of flight:
- Use the formula: distance = velocity * time,
where distance is the horizontal distance, velocity is the horizontal component of the minimum speed, and time is the time of flight.
- The horizontal component of the minimum speed can be calculated as follows:
horizontal velocity = v * cos(θ).
- Plug in the values into the equation and solve for the distance.
By following these steps, you can find the minimum speed required for the hammer to clear the edge of the well and the horizontal distance it will travel before landing outside the well.