Question

consider the diagram at the right in answering the next three questions. five locations along a roller coaster track are shown. assume that there are negligible friction and air resistance forces acting upon the coaster car.

1. rank the five locations in order of increasing tme (smallest to largest tme). use < and or = signs between the blanks.
2. rank the five locations in order of increasing pe (smallest to largest pe). use < and or = signs between the blanks.
3. rank the five locations in order of increasing ke (smallest to largest ke). use < and or = signs between the blanks.

Explanation:

Step1: Recall TME, PE, KE Relationships

Total Mechanical Energy (TME) is conserved (no friction/air resistance), so \( TME = PE + KE \) and \( TME_A = TME_B = TME_C = TME_D = TME_E \).
Potential Energy (PE) depends on height: \( PE = mgh \), so higher height means higher PE.
Kinetic Energy (KE) is \( KE = \frac{1}{2}mv^2 \), and \( KE = TME - PE \), so lower PE means higher KE (since TME is constant).

Step2: Analyze TME (Question 7)

Since TME is conserved (no non - conservative forces), all locations have equal TME. So \( A = B = C = D = E \).

Step3: Analyze PE (Question 8)

From the diagram, height order (lowest to highest): \( B \) (lowest) \( < D < E < C < A \) (highest, since \( A \) is the top of the first hill, \( C \) is top of loop, \( E \) is top of smaller hill, \( D \) is top of smaller loop, \( B \) is at ground level). So \( PE \) order (smallest to largest) is \( B < D < E < C < A \).

Step4: Analyze KE (Question 9)

Since \( KE = TME - PE \) and TME is constant, lower \( PE \) means higher \( KE \). So reverse the \( PE \) order. \( KE \) order (smallest to largest) is \( A < C < E < D < B \) (since \( A \) has highest \( PE \), so lowest \( KE \); \( B \) has lowest \( PE \), so highest \( KE \)).

Answer:

s:

1. \( A = B = C = D = E \)
2. \( B < D < E < C < A \)
3. \( A < C < E < D < B \)