Question

energy: potential vs kinetic, energy types and transformation, heat transfer

1. use the roller coaster to answer the following questions.

describe the energy at each point. label each point’s
potential and kinetic energy as increasing, decreasing,
highest, lowest, or medium.

a	b	c	d
ke:	ke:	ke:	ke:

a. at what point is kinetic energy increasing? ______
b. at any point on the rollercoaster, the total amount of energy is
(increasing / decreasing / the same)

Explanation:

Step 1: Recall the formulas for PE and KE

• Gravitational Potential Energy: \( PE = mgh \), where \( m \) is mass, \( g \) is acceleration due to gravity, and \( h \) is height.
• Kinetic Energy: \( KE=\frac{1}{2}mv^{2} \), where \( m \) is mass and \( v \) is velocity.

Step 2: Analyze Point A

• Height at A is maximum. Using \( PE = mgh \), maximum \( h \) gives maximum PE.
• Velocity at A is minimum (starts from rest or has minimum speed at the top), so using \( KE=\frac{1}{2}mv^{2} \), minimum \( v \) gives minimum KE.

Step 3: Analyze Point B

• As the roller - coaster moves from A to B, height \( h \) decreases. So from \( PE = mgh \), PE decreases.
• Velocity \( v \) increases as it moves down, so from \( KE=\frac{1}{2}mv^{2} \), KE increases.

Step 4: Analyze Point C

• Height at C is minimum. Using \( PE = mgh \), minimum \( h \) gives minimum PE.
• Velocity at C is maximum (since it has accelerated down from A), so from \( KE=\frac{1}{2}mv^{2} \), maximum \( v \) gives maximum KE.

Step 5: Analyze Point D

• Height at D is between A and C, so \( h \) is medium. Thus, \( PE = mgh \) is medium.
• Velocity at D is between A and C (slower than at C, faster than at A), so \( KE=\frac{1}{2}mv^{2} \) is medium.

Step 6: Analyze Part a (Kinetic Energy Increasing)

• As the roller - coaster moves from A to C, at point B, it is in the downward motion where velocity is increasing (because height is decreasing, PE is converting to KE). So KE is increasing at B.

Step 7: Analyze Part b (Total Energy)

• By the law of conservation of mechanical energy, \( PE + KE=\text{constant} \) (in the absence of non - conservative forces). So the total energy (sum of PE and KE) remains the same at any point on the roller - coaster.

Answer:

For the table (describing energy at each point):

Point A:

• PE: highest (since it's at the topmost point, maximum height means maximum gravitational potential energy, \( PE = mgh \), so as height \( h \) is maximum, PE is highest)
• KE: lowest (velocity is minimum or zero at the top, \( KE=\frac{1}{2}mv^{2} \), so KE is lowest)

Point B:

• PE: decreasing (moving down, height \( h \) decreases, so \( PE = mgh \) decreases)
• KE: increasing (velocity increases as it moves down, \( KE=\frac{1}{2}mv^{2} \), so KE increases)

Point C:

• PE: lowest (at the lowest point, height \( h \) is minimum, so \( PE = mgh \) is lowest)
• KE: highest (velocity is maximum at the lowest point, \( KE=\frac{1}{2}mv^{2} \), so KE is highest)

Point D:

• PE: medium (height is between A and C, so \( PE = mgh \) is medium)
• KE: medium (velocity is between A and C, so \( KE=\frac{1}{2}mv^{2} \) is medium)

Part a:

• At point B, kinetic energy is increasing (as the roller - coaster moves down from A to C, at B it is in the process of gaining speed, so KE is increasing)

Part b:

• The total amount of energy is the same (by the law of conservation of mechanical energy, in the absence of non - conservative forces like friction, the sum of potential energy and kinetic energy \( E = PE+KE \) remains constant. Even with friction, in an idealized roller - coaster problem for basic physics understanding, we assume conservation of mechanical energy, so total energy is the same at any point)