A conveyor belt is moving grain into a bin that is 4.80 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?
4.8/x = tan15°
To solve this problem, we can use the principles of projectile motion.
Let's break down the given information:
- The conveyor belt is inclined at an angle of 15.0°.
- The speed of the conveyor belt is 6.00 m/s.
- The grain does not slip on the conveyor belt.
- The grain needs to be dropped into a bin that is 4.80 m below the top of the conveyor belt.
We need to find the horizontal distance between the end of the conveyor belt and the bin.
First, let's find the time it takes for the grain to fall vertically from the conveyor belt to the bin. We can use the equation for vertical displacement:
y = ut + (1/2)gt^2
Where:
y = 4.80 m (vertical displacement)
u = 0 m/s (initial vertical velocity)
t = ? (time taken)
g = 9.8 m/s^2 (acceleration due to gravity)
Since the initial vertical velocity is 0 m/s (the grain is dropped), the equation simplifies to:
y = (1/2)gt^2
Solving for t:
4.80 = (1/2)(9.8)t^2
9.60 = 9.8t^2
t^2 = 9.60/9.8
t^2 = 0.9796
t = √0.9796
t ≈ 0.99 seconds
Now that we know it takes approximately 0.99 seconds for the grain to fall vertically, we can find the horizontal distance traveled by the grain in that time. We can use the equation for horizontal distance:
x = vt
Where:
x = ? (horizontal distance traveled)
v = 6.00 m/s (horizontal velocity)
t = 0.99 seconds (time taken)
Solving for x:
x = (6.00)(0.99)
x ≈ 5.94 meters
Therefore, the horizontal distance between the end of the conveyor belt and the bin must be approximately 5.94 meters.
To find the horizontal distance between the end of the conveyor belt and the bin, we can use trigonometry.
Given:
- Height of the bin above the conveyor belt: 4.80 m
- Inclination angle of the conveyor belt: 15.0°
- Speed of the conveyor belt: 6.00 m/s
We need to find the horizontal distance, which we'll call "d."
Using trigonometry, we can write:
sin(angle) = opposite/hypotenuse
Since the opposite side is the vertical distance (4.80 m) and the hypotenuse is the speed of the conveyor belt (6.00 m/s), we have:
sin(15.0°) = 4.80 m / 6.00 m/s
Rearranging the equation, we get:
4.80 m = sin(15.0°) * 6.00 m/s
Now, solve for the horizontal distance (d):
d = (4.80 m) / sin(15.0°)
Calculating this using a calculator, we find:
d ≈ 18.2 m
Therefore, the horizontal distance between the end of the conveyor belt and the bin must be approximately 18.2 meters.