Question

A conveyor belt is used to move sand from one place to another in a factory. The conveyor is tilted at an angle of \(14.0^{\circ}\) from the horizontal and the sand is moved without slipping at the rate of \(7.00 \mathrm{~m} / \mathrm{s}\). The sand is collected in a big drum \(3.00 \mathrm{~m}\) below the end of the conveyor belt. Determine the horizontal distance between the end of the conveyor belt and the middle of the collecting drum.

Short answer

Answer: The horizontal distance between the end of the conveyor belt and the middle of the collecting drum is approximately 5.29 meters.

Step-by-step solution

Calculate the time it takes for the sand to fall

We can use the following kinematic equation to find the time it takes for the sand to fall from the end of the conveyor belt to the middle of the drum: $$y = v_{0y} t + \frac{1}{2} a t^2$$ where \(y\) is the vertical displacement, \(v_{0y}\) is the initial vertical velocity, \(t\) is the time, and \(a\) is the vertical acceleration due to gravity. Since the sand falls freely from the end of the conveyor belt, we can assume that the initial vertical velocity \(v_{0y} = 0\), and the acceleration due to gravity \(a = -9.81 \mathrm{~m/s^2}\). We can rearrange the equation to solve for \(t\): $$t = \sqrt{\frac{2y}{a}}$$ Plug in the given value for \(y = 3.00 \mathrm{~m}\) and \(a = -9.81 \mathrm{~m/s^2}\): $$t = \sqrt{\frac{2 \times 3.00}{-9.81}} \approx 0.779 \mathrm{~s}$$

Calculate the horizontal velocity

We can calculate the horizontal component of the sand's velocity by using the angle of the conveyor belt and the given speed: $$v_x = v \cos{\theta}$$ where \(v\) is the speed of the sand on the conveyor belt and \(\theta\) is the angle between the conveyor belt and the horizontal. Plug in the given values for \(v = 7.00 \mathrm{~m/s}\) and \(\theta = 14.0^{\circ}\) (convert to radians): $$v_x = 7.00 \cos{14.0^\circ} = 7.00 \cos{0.244} \approx 6.79 \mathrm{~m/s}$$

Calculate the horizontal distance

We now have the time it takes for the sand to fall and the horizontal component of its velocity. To find the horizontal distance between the end of the conveyor belt and the middle of the collecting drum, we can use the following equation: $$x = v_x t$$ Plug in the values we calculated for \(v_x = 6.79 \mathrm{~m/s}\) and \(t = 0.779 \mathrm{~s}\): $$x = 6.79 \times 0.779 \approx 5.29 \mathrm{~m}$$ Therefore, the horizontal distance between the end of the conveyor belt and the middle of the collecting drum is approximately 5.29 meters.

Key concepts

Kinematics

Kinematics is a branch of physics that deals with the motion of objects without considering the forces that cause the motion. In this exercise, kinematics helps us analyze how the sand moves when it falls off the conveyor belt.

Key elements used in this problem include:

• Vertical motion: The sand's vertical motion is defined by its fall due to gravity. The kinematic equation for vertical displacement, \(y = v_{0y} t + \frac{1}{2} a t^2\), tells us how far the sand falls under gravity's influence.
• Time calculation: By manipulating the equation, we find the time, \(t\), it takes for the sand to reach the drum. The formula \(t = \sqrt{\frac{2y}{a}}\) helps us calculate this time.

Kinematics allows us to dissect motion into separate motions (horizontal and vertical) and solve each independently, providing a clear understanding of the whole motion.

Projectile Motion

Projectile motion describes the curved path that an object follows when it is thrown or propelled near the Earth's surface. The exercise involves analyzing how the sand moves through the air after leaving the conveyor belt.

In this case:

• Initial speed: The sand starts with a speed of \(7.00 \mathrm{~m/s}\), with both horizontal and vertical components.
• Horizontal component: This is critical for calculating how far the sand travels horizontally before hitting the drum. This action traces a trajectory typical of projectile motion.
• Gravity's role: Acting as a constant downward force, it influences the sand's vertical motion and ultimately its landing-point.

Understanding projectile motion allows us to predict where and when the sand will land.

Trigonometry

Trigonometry involves relationships between the angles and sides of triangles. In this problem, we use it to break down the velocity of the sand into horizontal and vertical components.

Here's how it works:

• Angle of the belt: The angle \(14.0^{\circ}\) helps determine how much of the sand's velocity points horizontally versus vertically.
• Calculating components: Using \(v_x = v \cos{\theta}\), we find the horizontal velocity, where \(\theta\) is the angle between the conveyor and horizontal.

Trigonometry is essential to understanding the direction and magnitude of movement, simplifying complex motion into simpler parts that can be more easily calculated.