Question

1. fill in the blanks in the following sentence: an object starts from rest with a potential energy of 600 j and free - falls towards the ground. after it has fallen to a height of one - fourth of its original height, its total mechanical energy is ____ j, its potential energy is __ j, and its kinetic energy is ____ j.

Explanation:

Step1: Recall Conservation of Mechanical Energy

In a free - fall (neglecting air resistance), the total mechanical energy (sum of potential energy \(PE\) and kinetic energy \(KE\)) of an object is conserved. The formula for gravitational potential energy is \(PE = mgh\), where \(m\) is mass, \(g\) is acceleration due to gravity, and \(h\) is height. The total mechanical energy \(E = PE+KE\). Initially, the object starts from rest, so initial kinetic energy \(KE_i = 0\) and initial potential energy \(PE_i=600\ J\). Thus, initial total mechanical energy \(E_i=PE_i + KE_i=600\ J+0\ J = 600\ J\). Since mechanical energy is conserved, at any point during the fall, \(E = 600\ J\).

Step2: Calculate Potential Energy at New Height

Let the original height be \(h_1\), so initial potential energy \(PE_1=mgh_1 = 600\ J\). After falling, the new height \(h_2=\frac{1}{4}h_1\). The new potential energy \(PE_2=mgh_2\). Substitute \(h_2=\frac{1}{4}h_1\) into the formula for \(PE_2\): \(PE_2 = mg\times\frac{1}{4}h_1=\frac{1}{4}(mgh_1)\). Since \(mgh_1 = 600\ J\), then \(PE_2=\frac{600}{4}=150\ J\).

Step3: Calculate Kinetic Energy at New Height

Using the conservation of mechanical energy \(E = PE_2+KE_2\). We know \(E = 600\ J\) and \(PE_2 = 150\ J\). Rearranging the formula for kinetic energy: \(KE_2=E - PE_2\). Substitute the values: \(KE_2=600\ J-150\ J = 450\ J\).

Answer:

The total mechanical energy is \(\boldsymbol{600}\) J, its potential energy is \(\boldsymbol{150}\) J, and its kinetic energy is \(\boldsymbol{450}\) J.