A conveyor belt is moving grain into a bin that is 5.00 m below the top of the conveyor belt. The grain does not slip on the conveyor belt that is inclined at 15.0° and they move at a constant speed of 6.00 m/s. In order for the conveyor belt to get the grain into the bin, what must the horizontal distance between the end of the conveyor belt and the bin be?
draw the diagram and review the basic trig functions.
5/x = tan15°
Well, to calculate the horizontal distance between the end of the conveyor belt and the bin, we need to use a little bit of trigonometry humor.
Now, we know that the grain is moving at a constant speed of 6.00 m/s, so let's call that the "run" of our conveyor belt.
But we also have an incline of 15.0°, which means we have to account for the "rise" of the conveyor belt as well.
Now, if we think about it, the "rise" is the vertical distance between the end of the conveyor belt and the bin, which is 5.00 m. So now we have a right-angled triangle where the "rise" is the opposite side and the "run" is the adjacent side.
Using some trigonometry magic, we can use the tangent function to find the angle between the "rise" and the "run."
So tan(15.0°) = opposite/adjacent, which means tan(15.0°) = 5.00 m/horizontal distance.
Now, we just have to solve for the horizontal distance, which is equal to 5.00 m divided by tan(15.0°).
So the horizontal distance between the end of the conveyor belt and the bin is approximately 17.16 m.
Voila! Now you have the grain sliding into the bin with a touch of trigonometry humor.
To find the horizontal distance between the end of the conveyor belt and the bin, we can break down the problem into two components: the vertical distance and the horizontal distance.
First, let's calculate the vertical distance the grain needs to travel. We are given that the bin is 5.00 m below the top of the conveyor belt, which means the vertical component of the displacement is -5.00 m (negative because it is going downward).
Since the conveyor belt does not slip, we can assume that the time taken for the grain to reach the bin is the same as the time taken for the grain to travel horizontally. This is because the grain travels at a constant speed of 6.00 m/s, both horizontally and vertically.
Next, let's calculate the time taken for the grain to reach the bin. We can use the formula for vertical displacement under constant acceleration:
v = u + at
where v is the final velocity (0 m/s since the grain comes to rest at the bin), u is the initial velocity (6.00 m/s), a is the acceleration (due to gravity, -9.8 m/s^2), and t is the time taken.
We can rearrange the formula to solve for time:
t = (v - u) / a
t = (0 - 6.00) / (-9.8)
t ≈ 0.6122 s
Since the time taken to reach the bin horizontally is also 0.6122 s, we can calculate the horizontal distance using the formula:
s = ut
where s is the horizontal distance, u is the initial velocity (6.00 m/s), and t is the time taken (0.6122 s).
s = 6.00 x 0.6122
s ≈ 3.67 m
Therefore, the horizontal distance between the end of the conveyor belt and the bin must be approximately 3.67 m.