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.
7 rank the five locations in order of increasing tme (smallest to largest tme). use < and or = signs between the blanks.
8 rank the five locations in order of increasing pe (smallest to largest pe). use < and or = signs between the blanks.
9 rank the five locations in order of increasing ke (smallest to largest ke). use < and or = signs between the blanks.
no friction nor air resistance) to determine the kinetic track below. finally,

Explanation:

Question 7 (TME - Total Mechanical Energy)

Step1: Recall TME Conservation

In a system with negligible friction and air resistance, total mechanical energy (TME = KE + PE) is conserved. So TME at all locations should be equal.

Step2: Apply Conservation Law

Since TME is conserved, \( A = B = C = D = E \).

Step1: Relate PE to Height

Potential Energy (PE) in a gravitational field is \( PE = mgh \), so PE depends on height (\( h \)). Higher height means higher PE.

Step2: Analyze Heights

From the diagram: Height of \( B \) is lowest, then \( D \), then \( E \), then \( C \) and \( A \)? Wait, no—wait, \( A \) is a high point, \( B \) is low, \( C \) is top of loop, \( D \) is top of smaller loop, \( E \) is a peak. Wait, let's re - assess: \( B \) is at the bottom, so \( h_B \) is minimum. Then \( D \) (top of smaller loop) has lower height than \( E \)? Wait, no, the diagram: \( A \) is a high slope, \( B \) is bottom, \( C \) is top of a big loop, \( D \) top of small loop, \( E \) a peak. Let's order heights: \( h_B < h_D < h_E < h_C = h_A \)? Wait, no, maybe \( A \) and \( C \) and \( E \)? Wait, no, let's think again. PE is proportional to height. So the lowest height is \( B \), then \( D \) (top of small loop), then \( E \) (peak), then \( C \) (top of big loop) and \( A \) (the starting high point)? Wait, maybe \( h_B < h_D < h_E < h_C = h_A \)? Wait, no, perhaps \( A \) and \( C \) have the same height? Wait, the problem says "five locations": \( A, B, C, D, E \). Let's assume: \( B \) is at the lowest height, then \( D \) (top of small loop), then \( E \) (a peak), then \( C \) (top of big loop) and \( A \) (the high - left point). Wait, maybe \( h_B < h_D < h_E < h_C = h_A \)? So PE order: \( PE_B < PE_D < PE_E < PE_C = PE_A \) (since \( PE = mgh \), mass \( m \) and \( g \) are constant, so PE ∝ h).

Step3: Write the Order

So the order of increasing PE is \( B < D < E < C = A \) (or check the heights again: maybe \( A \) is same as \( C \), \( D \) lower than \( E \), \( B \) lowest).

Step1: Recall KE - PE Relationship

Since \( TME = KE + PE \) and \( TME \) is conserved, \( KE = TME - PE \). So lower PE means higher KE, and vice - versa.

Step2: Use PE Order to Find KE Order

From Question 8, the order of PE (increasing) is \( B < D < E < C = A \). So for KE, since \( KE = TME - PE \), the order of KE (increasing) will be the reverse of PE (because as PE increases, KE decreases). So \( KE_A = KE_C < KE_E < KE_D < KE_B \) (since \( PE_A = PE_C \) (highest PE), so \( KE_A = KE_C \) (lowest KE), then \( PE_E \) is next, so \( KE_E \) is next, then \( PE_D \), so \( KE_D \), then \( PE_B \) (lowest PE), so \( KE_B \) (highest KE)).

Step3: Write the Order

So the order of increasing KE is \( A = C < E < D < B \).

Answer:

\( A = B = C = D = E \)

Question 8 (PE - Potential Energy)