Kinetic and Potential Energy Questions - Free Printable

Let’s go step by step to solve each question. We’re dealing with a roller coaster car that starts at rest at point A, and we assume no friction (so energy is conserved — total mechanical energy = kinetic + potential stays the same).

We’re told:
- Mass of car = 710. kg
- Starts from rest at point A → so initial kinetic energy = 0
- Gravity g = 9.81 m/s²

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Question 1: At what point is the energy of the car equivalent to 0J?

Wait — this is tricky. The *total* energy isn’t zero anywhere because the car has mass and height or speed. But if they mean “where is both kinetic AND potential energy zero?” — that would be impossible unless the car is at ground level AND not moving.

But looking at the diagram, point A is where it starts from rest — but it’s elevated (since it goes up to B). So potential energy at A is NOT zero.

Actually — rethinking: maybe they mean “at which point is the kinetic energy 0J?” Since it starts from rest at A, and assuming it stops again somewhere? But in a frictionless system, it won’t stop unless it reaches the same height as A on the other side — which it doesn’t seem to.

Wait — look at the answer key says “Point A”. Why?

Because at Point A, the car is at rest → kinetic energy = 0. And if we define the reference point for potential energy (height = 0) at point A, then potential energy at A is also 0. So total energy at A = 0? That can’t be right because later it gains energy.

Actually — here’s the correct interpretation:

In physics problems like this, we often set the lowest point as h=0 for potential energy. But the problem doesn’t specify. However, since the car starts at rest at A, and if we take A as our reference point (h=0), then PE at A = 0, KE at A = 0 → total energy = 0? That would violate conservation of energy because later it has positive energy.

That doesn’t make sense.

Alternative idea: Maybe the question means “at which point is the kinetic energy 0J?” — because it starts from rest at A, and if there were another point where it momentarily stops, KE=0 there too. But in the diagram, only A is labeled as starting from rest.

Looking at the provided answer key: it says “Point A” — so likely they mean: at point A, since it starts from rest, kinetic energy is 0. And perhaps they are considering that as the reference point for potential energy too? Or maybe it’s a trick — total energy is never 0, but kinetic energy is 0 at A.

I think the intended answer is Point A, because that’s where the car is at rest → KE = 0. Even though PE may not be zero, maybe the question is poorly worded and they mean “kinetic energy is 0”.

But let’s check the context. Question 2 asks about highest potential energy — which is at B. So they are treating heights relative to some baseline.

Perhaps for question 1, they mean: at which point is the kinetic energy 0? Because total energy can’t be 0 unless everything is zero.

Given the answer key says “Point A”, I’ll go with that — assuming they mean kinetic energy is 0 at A (which it is, since it starts from rest).

✔ Final Answer for Q1: Point A

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Question 2: At what point(s) is potential energy the highest?

Potential energy depends on height: PE = mgh. Highest height → highest PE.

From the diagram, point B is the highest peak.

✔ Final Answer for Q2: Point B

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Question 3: Assuming frictionless, at what point(s) is kinetic energy the highest?

Kinetic energy is highest when potential energy is lowest (because total energy is conserved). Lowest points on the track are C and E — they look like valleys.

So KE is max at C and E.

✔ Final Answer for Q3: Point C and Point E

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Question 4: How does PE at B compare to PE at D?

B is higher than D → so PE at B > PE at D.

✔ Final Answer for Q4: The potential energy at point B is greater than point D.

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Question 5: How does KE at B compare to KE at G?

At B, the car is at high elevation → low speed → low KE.

At G, it’s lower than B → so more KE.

Also, since total energy is constant, and PE_B > PE_G → KE_B < KE_G.

✔ Final Answer for Q5: The kinetic energy at point B is less than point G.

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Now questions 6–9 require calculations.

Recall:
- Total mechanical energy is conserved (frictionless)
- At point A: starts from rest → KE_A = 0
- Let’s set PE = 0 at point A? Wait — but we don’t know height of A. Actually, we need to pick a reference.

Better approach: Since the car starts at rest at A, all its energy at any point comes from converting potential energy to kinetic energy as it falls.

But we don’t know height of A. However, for questions 6–9, they give heights relative to... probably sea level or arbitrary, but we can calculate differences.

Actually, for conservation of energy between two points:

Total energy at start (A) = PE_A + KE_A = PE_A + 0

At any other point X: PE_X + KE_X = PE_A

So KE_X = PE_A - PE_X = mg(h_A - h_X)

But we don’t know h_A.

Wait — for question 6: they say “Point B has a height of 45.0 m.” — but height relative to what? Probably relative to point A? Or absolute?

This is ambiguous. But in roller coaster problems, usually heights are given relative to the lowest point or to the starting point.

Looking at question 7: “Point B has height 45.0 m. What is speed at C?” — and C is a valley. If we assume that point C is at height 0 (lowest point), then we can compute.

But the diagram shows multiple valleys — C and E are both low, but maybe not same height? The problem doesn’t specify.

Actually, for calculation purposes, since they give specific heights for B, D, G, and ask for speeds or energies, we must assume that these heights are measured from a common reference — likely the ground or the lowest point.

Moreover, since the car starts at A, and A is not at height 0 (it’s below B), we need to know height of A.

Wait — here’s the key: at point A, the car is at rest. So total energy E_total = PE_A = m * g * h_A

At any point, PE + KE = E_total

For example, at point B: PE_B = m*g*h_B, KE_B = ?

But we don’t know h_A.

Unless... perhaps for questions 6–9, they want us to use the height values given directly, and assume that the potential energy is calculated from those heights, and kinetic energy derived from conservation.

But without knowing h_A, we can’t find absolute PE.

Unless — for question 6: “What is the potential energy of the car at B?” — they just want PE_B = m*g*h_B, using h_B = 45.0 m, regardless of reference.

Similarly for others.

And for speed at C: if we assume that at C, height is 0 (since it’s a valley and often taken as reference), then all PE at B converts to KE at C.

That makes sense.

Let me check the answer key:

Q6: 313,000 J → let’s calculate: m=710, g=9.81, h=45.0

PE = 710 * 9.81 * 45.0 = ?

First, 710 * 9.81 = let's compute:

700 * 9.81 = 6867

10 * 9.81 = 98.1

Total = 6867 + 98.1 = 6965.1

Then * 45 = 6965.1 * 45

6965.1 * 40 = 278,604

6965.1 * 5 = 34,825.5

Sum = 278,604 + 34,825.5 = 313,429.5 ≈ 313,000 J (rounded to 3 significant figures? 710 has 3, 45.0 has 3, 9.81 has 3 → so answer should have 3 sig figs → 313,000 is 3.13 × 10^5, which is 3 sig figs. Yes.

So for Q6, they just want PE at B = mgh_B, with h_B=45.0m.

Similarly, for Q7: speed at C. If we assume that point C is at height 0 (reference), then all PE at B converts to KE at C.

So KE_C = PE_B = 313,429.5 J

KE = (1/2)mv² → v = sqrt(2*KE/m)

v = sqrt(2 * 313429.5 / 710)

First, 2 * 313429.5 = 626,859

Divide by 710: 626859 / 710 ≈ ?

710 * 883 = 710*800=568,000; 710*83=58,930; total 626,930 → close to 626,859

626,930 - 626,859 = 71, so approximately 883 - 0.1 = 882.9

Better calculate:

626859 ÷ 710

710 * 882 = 710*800=568,000; 710*82=58,220; total 626,220

Subtract: 626,859 - 626,220 = 639

So 882 + 639/710 ≈ 882 + 0.9 = 882.9

So v = sqrt(882.9) ≈ ?

sqrt(900)=30, sqrt(841)=29, so around 29.7

29.7^2 = 882.09 → yes, very close.

So v ≈ 29.7 m/s → matches answer key.

So assumption: point C is at height 0.

Similarly, for other points, we need their heights.

But the problem doesn't give height of C, E, etc. Only B, D, G are given in later questions.

For Q7, they only give height of B, and imply C is at 0.

For Q8: Point D height 30.0 m. Speed at D?

Total energy at start: at A, but we don't know h_A.

Problem: if we set PE=0 at C (height 0), then at A, what is height? Not given.

But at point B, h=45.0m, and car started from rest at A.

If A is not at height 0, then total energy is mg*h_A.

At B, PE_B = mg*45.0, KE_B = total - PE_B = mg(h_A - 45.0)

But we don't know h_A.

However, for the car to reach B, h_A must be less than or equal to 45.0? No, it starts at A and goes up to B, so h_A < h_B.

Actually, since it starts from rest at A and goes up to B, that means A must be lower than B, and the car must have been pushed or something? But the problem says "starts from rest at point A", and then goes up to B — that implies that A is not the lowest point; perhaps there's a launch or something, but it's not specified.

This is confusing.

Alternative interpretation: perhaps "starts from rest at point A" means that at A, it is released, and A is at some height, and it rolls down, but in the diagram, from A it goes up to B — that would require initial push, but it says "from rest".

Looking back at the diagram description: "A 710. kg car travels along a roller coaster track. The car starts from rest at point A."

And the diagram shows A at the bottom left, then up to B. So if it starts from rest at A and goes up to B, that violates conservation of energy unless there's an external force.

That can't be.

Perhaps I misread the diagram. Let me visualize: typically, roller coasters start at a high point and go down. But here, it says starts from rest at A, and A is shown at the beginning of the track, which might be low, but then it goes up to B — that doesn't make sense for starting from rest.

Unless... perhaps point A is actually at a high point? But the diagram labels A at the start, and B is the first hill.

I think there might be a mistake in my assumption.

Let me read carefully: "The car starts from rest at point A." and the diagram has A at the left end, then curve up to B.

In standard roller coaster problems, the car is pulled up to the first hill and released from rest there. But here, it says starts from rest at A, and A is not the top.

Perhaps A is the top? But the label is at the bottom.

Another possibility: maybe "starts from rest at point A" means that at the beginning of the motion, it is at A with zero velocity, and A is at a certain height, and then it moves to B which is higher — but that would require work done, which isn't mentioned.

This is problematic.

But looking at the answer key for Q10: "Why is B the highest point? The first hill of a roller coaster is the highest because that is when total potential energy is the greatest."

And for Q1, answer is Point A for 0J energy — which suggests that at A, energy is 0, so perhaps they set PE=0 at A, and since it starts from rest, KE=0, so total energy=0.

Then at B, PE_B = mgh_B, but if total energy is 0, then KE_B = -PE_B, which is negative — impossible.

That doesn't work.

Unless they mean that the potential energy is defined such that at A, PE=0, and since it starts from rest, total energy=0, but then when it goes up to B, PE increases, so KE must be negative — absurd.

I think the only logical way is to assume that "starts from rest at point A" means that A is the highest point, but the diagram shows B as higher.

Perhaps there's a typo in the problem or diagram.

Let's look at the answer key for Q7: speed at C is 29.7 m/s, which we calculated assuming that at B, PE = mgh_B, and at C, h=0, so KE_C = PE_B.

But if the car starts from rest at A, and A is not at h=0, then total energy is mg*h_A, not mg*h_B.

Unless h_A = h_B? But that can't be.

Another idea: perhaps "starts from rest at point A" is a mistake, and it should be "released from rest at point B" or something.

But the problem explicitly says "at point A".

Let's calculate what h_A must be for the car to reach B.

Suppose at A, height h_A, velocity 0.

At B, height h_B = 45.0 m, velocity v_B.

Conservation: mg*h_A = mg*h_B + (1/2)m*v_B^2

So v_B^2 = 2g(h_A - h_B)

For v_B to be real, h_A > h_B, but in the diagram, B is higher than A, so h_B > h_A, so v_B^2 negative — impossible.

Therefore, the only way this makes sense is if the car is not starting from rest at A in the sense of being released, but rather, A is the point where it is launched or something, but the problem says "from rest".

Perhaps "starts from rest at point A" means that at the very beginning, before any motion, it is at A with v=0, and then it is given a push or the track is designed so that it can go up, but that requires external energy.

I think for the sake of solving the problem as per the answer key, we must assume that the total mechanical energy is based on the height at B or something.

Notice that in Q6, they ask for PE at B with h=45.0m, and get 313,000 J, which is m*g*45.0.

In Q7, speed at C is 29.7 m/s, which is sqrt(2*g*45.0) = sqrt(2*9.81*45) = sqrt(882.9) = 29.7, which suggests that they are assuming that at C, all the PE from B is converted to KE, implying that C is at height 0, and B is at 45.0m, and the car had enough energy to reach B from A, but for the calculation of speed at C, they are using the drop from B to C.

But how did it get to B if it started from rest at A which is lower?

Perhaps for questions 6-9, they are considering the energy at B as the total energy, ignoring A.

Or perhaps "starts from rest at point A" is irrelevant for 6-9, and we should use the given heights directly.

Let's try that.

For Q6: PE at B = m*g*h_B = 710 * 9.81 * 45.0 = 313,429.5 J ≈ 313,000 J (3 sig figs)

For Q7: speed at C. If we assume that point C is at height 0, and the car falls from B to C, then loss in PE = gain in KE.

So mg*45.0 = (1/2)m*v^2 → v = sqrt(2*g*45.0) = sqrt(2*9.81*45) = sqrt(882.9) = 29.71 m/s ≈ 29.7 m/s

For Q8: Point D has height 30.0 m. Speed at D?

If total energy is conserved, and we take the total energy as the PE at B (since at B, if it were at rest, but it's not, but in this case, at B, the car has some KE and PE.

This is messy.

Assume that the total mechanical energy is constant, and at point B, we can find the total energy if we know the speed, but we don't.

From the start: at A, v=0, h=h_A.

At B, h=45.0m, v=v_B.

But unknowns.

Perhaps for the purpose of this problem, since they give heights for B, D, G, and ask for speeds, and the answer key gives specific numbers, we can assume that the reference point for height is such that at the lowest points (C and E), h=0, and at A, h is some value, but for calculations involving B, D, G, we can use the given heights and conservation from B or from A.

Let's calculate the speed at D.

Suppose we take total energy E = PE at B + KE at B.

But we don't know KE at B.

From the diagram, at B, it's a peak, so if it's smooth, the speed at B is not zero, but minimum for that section.

This is complicated.

Another approach: perhaps "starts from rest at point A" and A is at height 0, and B is at 45.0m, but then to go from A to B, it needs initial kinetic energy, but it starts from rest, so impossible.

I think there might be a mistake in the problem setup, but since the answer key is provided, and for Q7 they got 29.7 m/s by assuming drop from B to C with h_B=45.0m and h_C=0, similarly for others, we'll proceed with that assumption for calculations.

So for Q8: Point D has height 30.0 m. What is speed at D?

If we assume that the total energy is the same as at B, but at B, if the car is at height 45.0m, and if we knew its speed, but we don't.

Unless at B, since it's a maximum, and if we assume that the car was released from a higher point, but it's not specified.

Perhaps for the entire ride, the total energy is determined by the initial condition at A.

Let's define h=0 at point C (lowest point).

Then at B, h_B = 45.0 m (given).

At A, what is h_A? From the diagram, A is before B, and it's going up, so h_A < h_B.

But if it starts from rest at A, then total energy E = mg*h_A.

At B, PE_B = mg*45.0, KE_B = E - PE_B = mg(h_A - 45.0)

For KE_B >=0, h_A >=45.0, but if h_A >45.0, then A is higher than B, but in the diagram, B is the first hill, so probably h_A < h_B, contradiction.

Unless the diagram is misleading, and A is actually at a high point.

Perhaps "point A" is at the top of the first hill, but the label is placed at the bottom by mistake.

In many diagrams, the starting point is labeled at the top.

Let me check the user's image description: "A 710. kg car travels along a roller coaster track. The car starts from rest at point A." and the diagram has A at the left end, which is likely the start, and then it goes up to B.

But in standard problems, the car is at rest at the top of the first hill.

Perhaps for this problem, we should assume that point A is at height 0, and the car is given initial energy, but it says "from rest".

I think to resolve this, let's look at Q9: Point G has height 20.0 m. How much kinetic energy at G? Answer key says 22.1 m/s — wait, that's speed, not energy. The question asks for kinetic energy, but answer is in m/s? That must be a typo in the key or in my reading.

Let's see: "9) Point G has a height of 20.0 m. How much kinetic energy does the car have at point G? 22.1 m/s"

22.1 m/s is speed, not energy. Probably a mistake in the key; it should be the speed, or they meant to ask for speed.

In the text: "How much kinetic energy" but answer is "22.1 m/s" — inconsistency.

Perhaps it's a typo, and it should be "what is the speed".

Because 22.1 m/s is a speed.

Similarly, in Q7 and Q8, they ask for speed.

So likely for Q9, it should be "what is the speed at G".

And the answer is 22.1 m/s.

So let's assume that.

For Q9: speed at G, h_G=20.0m.

If we assume that total energy is mg*45.0 (as if released from B with v=0, but it's not), then at G, PE_G = mg*20.0, KE_G = mg*45.0 - mg*20.0 = mg*25.0

Then v = sqrt(2*KE/m) = sqrt(2*g*25.0) = sqrt(2*9.81*25) = sqrt(490.5) = 22.15 m/s ≈ 22.1 m/s — matches.

Similarly for Q8: at D, h_D=30.0m, so if total energy corresponds to h=45.0m, then KE_D = mg*(45.0 - 30.0) = mg*15.0

v = sqrt(2*g*15.0) = sqrt(2*9.81*15) = sqrt(294.3) = 17.15 m/s ≈ 17.1 m/s — matches answer key.

For Q7: at C, h_C=0 (assumed), KE_C = mg*45.0, v= sqrt(2*g*45.0) =29.7 m/s.

So the assumption is: the total mechanical energy is equal to the potential energy at height 45.0 m, as if the car were released from rest at a height of 45.0 m.

But the problem says it starts from rest at point A, not at B.

However, for the calculations to match, we must assume that the effective "starting height" for energy purposes is 45.0 m, i.e., point B is where it would have been released from rest, but the problem says A.

Perhaps in the diagram, point A is at the same height as B, but that doesn't make sense.

Another possibility: "starts from rest at point A" and A is at height 45.0 m, but the diagram labels B as the top, so maybe A and B are the same point? Unlikely.

I think for the sake of solving, we'll proceed with the calculation as per the answer key, assuming that the total energy is mg*45.0 J, corresponding to being at rest at height 45.0 m.

So for all calculations, total energy E = m * g * 45.0 = 710 * 9.81 * 45.0 = 313,429.5 J

Then at any point with height h, PE = m*g*h, KE = E - PE = m*g*(45.0 - h)

Then speed v = sqrt(2*KE/m) = sqrt(2*g*(45.0 - h))

Now let's solve each.

Q6: PE at B, h_B=45.0 m

PE_B = m*g*h_B = 710 * 9.81 * 45.0 = 313,429.5 J

Round to correct sig figs: 710 has 3, 9.81 has 3, 45.0 has 3, so product should have 3 sig figs.

313,429.5 → 313,000 J (since 3.13 × 10^5)

Yes.

Q7: speed at C. Assume h_C = 0 m (lowest point)

KE_C = m*g*(45.0 - 0) = m*g*45.0

v_C = sqrt(2 * g * 45.0) = sqrt(2 * 9.81 * 45.0) = sqrt(882.9) = 29.712... ≈ 29.7 m/s (3 sig figs)

Q8: speed at D, h_D=30.0 m

v_D = sqrt(2 * g * (45.0 - 30.0)) = sqrt(2 * 9.81 * 15.0) = sqrt(294.3) = 17.155... ≈ 17.1 m/s (3 sig figs)

Q9: speed at G, h_G=20.0 m (assuming they meant speed, not energy)

v_G = sqrt(2 * g * (45.0 - 20.0)) = sqrt(2 * 9.81 * 25.0) = sqrt(490.5) = 22.147... ≈ 22.1 m/s (3 sig figs)

Q10: Why is B the highest point?

As per answer key: "The first hill of a roller coaster is the highest because that is when total potential energy is the greatest."

In a frictionless system, the car cannot go higher than its starting height. Since it starts from rest at A, and if A is at a certain height, B cannot be higher than A. But in this case, if we assume that the starting point for energy is effectively at 45.0 m, then B is at that height, and subsequent hills are lower because energy is conserved, so it can't exceed the initial potential energy.

But the problem says it starts from rest at A, not at B.

Perhaps in the context, "the first hill" refers to B, and it's the highest because that's where the maximum potential energy occurs during the ride, and since no additional energy is added, it can't go higher later.

So the reason is that after the first hill, the car loses some energy to friction in real life, but here it's frictionless, so theoretically it could return to the same height, but in practice, roller coasters are designed with the first hill highest to ensure it completes the track.

But for this problem, since it's frictionless, and if it starts from rest at A, and A is lower than B, it shouldn't reach B, but we're forcing it.

I think for Q10, the intended answer is that B is the highest point because it's the first hill, and in roller coasters, the first hill is always the tallest to provide enough energy for the rest of the ride.

So we'll go with that.

Now to summarize answers:

1) Point A
2) Point B
3) Point C and Point E
4) The potential energy at point B is greater than point D.
5) The kinetic energy at point B is less than point G.
6) 313,000 J
7) 29.7 m/s
8) 17.1 m/s
9) 22.1 m/s (assuming they meant speed)
10) The first hill of a roller coaster is the highest because that is when total potential energy is the greatest.

For Q9, the question asks for kinetic energy, but the answer is given as speed. Probably a typo in the problem or key. Since the key says 22.1 m/s, and it matches the speed calculation, I'll report speed.

If they really want kinetic energy at G, then KE_G = m*g*(45.0 - 20.0) = 710 * 9.81 * 25.0 = let's calculate: 710*9.81=6965.1, times 25=174,127.5 J ≈ 174,000 J, but the key says 22.1 m/s, so likely they meant speed.

So I'll go with speed for Q9.

Final Answers:

1) Point A
2) Point B
3) Point C and Point E
4) The potential energy at point B is greater than point D.
5) The kinetic energy at point B is less than point G.
6) 313,000 J
7) 29.7 m/s
8) 17.1 m/s
9) 22.1 m/s
10) The first hill of a roller coaster is the highest because that is when total potential energy is the greatest.

Note: For questions 6-9, we assumed that the total mechanical energy corresponds to the potential energy at a height of 45.0 m, as if the car were released from rest at that height, even though the problem states it starts from rest at point A. This assumption is necessary to match the provided answer key and make the calculations consistent.