A freight company uses a compressed spring to shoot 2.0 kg packages up a 1.0-m-high frictionless ramp into a truck, as FIGURE P10.52 shows. The spring constant is 500 N/m and the spring is compressed 30 cm. What is the speed of the package when it reaches the truck?

검증된 단계별 안내

Step 1: Identify the energy transformations in the system. The spring's potential energy is converted into the package's kinetic energy and gravitational potential energy as it moves up the ramp. Use the principle of conservation of energy to solve the problem.

Step 2: Write the formula for the spring's potential energy: \( U_s = \frac{1}{2} k x^2 \), where \( k \) is the spring constant (500 N/m) and \( x \) is the compression distance (0.30 m). Substitute these values to calculate the spring's potential energy.

Step 3: Write the formula for gravitational potential energy: \( U_g = m g h \), where \( m \) is the mass of the package (2.0 kg), \( g \) is the acceleration due to gravity (9.8 m/s²), and \( h \) is the height of the ramp (1.0 m). Substitute these values to calculate the gravitational potential energy.

Step 4: Apply the conservation of energy principle: \( U_s = K + U_g \), where \( K \) is the kinetic energy of the package at the top of the ramp. Rearrange the equation to solve for \( K \): \( K = U_s - U_g \). Use the values calculated in Steps 2 and 3.

Step 5: Use the formula for kinetic energy: \( K = \frac{1}{2} m v^2 \), where \( v \) is the speed of the package. Rearrange the equation to solve for \( v \): \( v = \sqrt{\frac{2K}{m}} \). Substitute the value of \( K \) from Step 4 and the mass of the package to find the speed.

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주요 개념

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

Potential Energy of a Spring

The potential energy stored in a compressed spring is given by the formula PE_spring = (1/2)kx^2, where k is the spring constant and x is the compression distance. In this scenario, the spring constant is 500 N/m and the spring is compressed by 0.3 m, allowing us to calculate the energy available to propel the package.

Conservation of Energy

The principle of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another. In this problem, the potential energy stored in the spring is converted into kinetic energy as the package moves up the ramp, allowing us to relate the energies to find the speed of the package.

Kinetic Energy

Kinetic energy (KE) is the energy of an object due to its motion, calculated using the formula KE = (1/2)mv^2, where m is the mass and v is the velocity. To find the speed of the package when it reaches the truck, we will equate the kinetic energy to the potential energy converted from the spring and the gravitational potential energy gained while moving up the ramp.