Doug uses a 25 N horizontal force to push a 5.0 kg crate up a 2.0-m-high, 20° frictionless slope. What is the speed of the crate at the top of the slope?

검증된 단계별 안내

Identify the forces acting on the crate: The applied force (25 N), the gravitational force, and the normal force. Since the slope is frictionless, there is no frictional force.

Break the gravitational force into components parallel and perpendicular to the slope. The parallel component is given by \( F_{\text{gravity, parallel}} = m g \sin(\theta) \), where \( m = 5.0 \ \text{kg} \), \( g = 9.8 \ \text{m/s}^2 \), and \( \theta = 20^\circ \).

Determine the net force acting along the slope: \( F_{\text{net}} = F_{\text{applied}} - F_{\text{gravity, parallel}} \). Use the values of \( F_{\text{applied}} = 25 \ \text{N} \) and the calculated \( F_{\text{gravity, parallel}} \).

Use Newton's second law to find the acceleration of the crate along the slope: \( a = \frac{F_{\text{net}}}{m} \). Substitute the net force and the mass of the crate to calculate \( a \).

Apply the kinematic equation \( v^2 = u^2 + 2 a d \) to find the speed \( v \) of the crate at the top of the slope. Here, \( u = 0 \ \text{m/s} \) (initial speed), \( a \) is the acceleration calculated in the previous step, and \( d \) is the distance along the slope, which can be found using trigonometry: \( d = \frac{h}{\sin(\theta)} \), where \( h = 2.0 \ \text{m} \).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Newton's Second Law of Motion

Newton's Second Law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. This principle is crucial for analyzing the motion of the crate as it is pushed up the slope, allowing us to calculate the net force and resulting acceleration.

Work-Energy Principle

The Work-Energy Principle asserts that the work done on an object is equal to the change in its kinetic energy. In this scenario, the work done by Doug's force will contribute to the crate's kinetic energy as it moves up the slope, enabling us to determine its speed at the top.

Kinematics

Kinematics is the branch of mechanics that deals with the motion of objects without considering the forces that cause the motion. Understanding kinematic equations is essential for calculating the final speed of the crate after it has been pushed up the slope, given its initial conditions and the work done on it.