How is momentum conserved in a collision? A momentum and energy quiz

Chapter: Defining Momentum If you try to stop a tennis ball rolling toward you at a walking pace, you can do it with just a finger. However, if a heavy bowling ball rolls toward you at that same walking speed, stopping it requires a much firmer grip and more effort. Even though both balls are moving at the same speed, the bowling ball has more 'oomph' or 'quantity of motion' because it is heavier. In physics, we call this quantity of motion momentum, and it depends on two specific factors: how heavy the object is and how fast it is moving. This formula tells us that momentum isn't just about being heavy; it's also about being fast. A very light object can have the same momentum as a very heavy one if it moves fast enough. No. Because the mountain's velocity is zero, its momentum ($m \times 0$) is also zero. The mosquito has more momentum because it is actually moving. So far, we have only talked about the magnitude—the 'how much'—of momentum. But there is another vital piece to the puzzle. Momentum is a vector quantity. This means that the direction in which an object moves is just as important as how fast it is going. If you are catching a ball, it matters if it is coming toward you or moving away from you. The direction defines the momentum's effect. Because direction matters, physicists often use a negative sign to show an object moving in the opposite direction. If 'Right' is positive (+10 kg·m/s), then 'Left' would be negative (-10 kg·m/s). Understanding momentum is the first step in predicting what happens when objects crash, push, or pull on each other. In the next chapters, we will see how applying a force over time can change this momentum—a concept known as impulse. We already use 'm' for mass. Scientists chose 'p' from the Latin word *petere*, which means 'to go' or 'to seek', to avoid confusion and represent the 'progress' of an object. No. Momentum is a property an object *has* while moving ($mv$). Force is something *applied* to an object to change its motion ($ma$). However, they are related: a force applied over time is what changes an object's momentum. Yes, but only in terms of direction. A negative sign doesn't mean the object has 'less than zero' motion; it just means it is moving in the opposite direction of what you defined as 'positive'. Not directly. Momentum only depends on mass and velocity. However, gravity can change an object's velocity (like a falling ball), which in turn changes its momentum. As the car slows down, its velocity decreases. Since mass stays the same, its momentum gradually drops to zero. The 'missing' momentum is transferred to the Earth through friction. Chapter: Impulse and Change A golf ball sitting on a tee requires a specific interaction to transform from a stationary object into a high-speed projectile. This combination of how hard you hit something and how long the contact lasts is a physical quantity called Impulse. In physics, Impulse is not just a sudden urge; it is the mathematical product of the force applied and the duration of that application. Since we multiply Force (Newtons) by Time (seconds), the standard unit for Impulse is the Newton-second ($N \cdot s$). Previously, we defined Momentum as the 'oomph' of a moving object, calculated by multiplying its mass and its velocity. There is a deep connection between the Impulse applied to an object and the resulting change in that object's Momentum. This derivation brings us to the Impulse-Momentum Relationship: the impulse delivered to an object is exactly equal to its change in momentum. Following through ensures the club stays in contact with the ball for the longest possible Time interval. Increasing time increases the total Impulse, which gives the ball a much higher final velocity. When a soccer player kicks a ball, they apply a large Force over a very short Time to create a massive change in momentum. This relationship works in reverse when we try to stop an object that is already moving. To stop a moving car, we must reduce its Momentum to zero. This requires a specific amount of Impulse. Since Impulse is Force multiplied by Time, we have a choice: we can use a huge force for a short time, or a small force for a long time. By extending the Time of an impact, we decrease the average Force acting on the object (or person) involved. Modern cars are designed with Crumple Zones specifically to exploit this principle of physics. You naturally use this physics principle when you 'give' with your hands while catching a fast-moving baseball. Bungee jumping works on the same logic; the stretchy cord ensures the jumper stops over several seconds rather than a lethal fraction of a second. Whether you are hitting a golf ball for maximum distance or designing a car for maximum safety, you are managing the delicate balance between Force, Time, and Momentum. We use Impulse to describe the *process* of the interaction (the force and time applied), while Change in Momentum describes the *result* of that interaction (how the object's mass and velocity were affected). It's like 'payment' vs. 'balance change' in a bank account. Yes! In real life, forces usually spike and then fade (like a bat hitting a ball). The theorem still works perfectly using the average force over the time interval, or by calculating the total area under a Force-Time graph. No. If a force is applied but the object doesn't move, it is likely being balanced by another force (like friction). If there is a *net* impulse, there must be a change in momentum, which implies a change in velocity. Seatbelts are designed to stretch slightly. While they feel firm, that small amount of stretch (and the way they distribute force over your torso) still takes much more time than hitting the rigid steering wheel or windshield directly. They are related but different. Impulse is Force acting over Time and changes momentum. Work is Force acting over Distance and changes kinetic energy. They are two different 'lenses' for looking at the same physical event. Chapter: Conservation of Momentum A cue ball glides across the smooth felt of a pool table and strikes a stationary 8-ball directly in the center. When the cue ball hits the 8-ball, the cue ball slows down significantly. It loses some of its momentum during the interaction. Simultaneously, the 8-ball, which was sitting perfectly still, suddenly bolts forward. It has gained exactly what the cue ball lost. If you add the 'After' values (3 + 7), you get 10. This is the same total amount we started with. To track this properly, we define a system. This is simply the collection of objects we choose to study together. Inside our pool ball system, the forces they exert on each other during the hit are called internal forces. It is an internal force because both the source and the receiver of the force are part of the system. The Law of Conservation of Momentum states that the total momentum of a system remains constant if no outside forces interfere. This means that while individual objects change their speed, the sum of all momentum stays exactly the same. This law is a direct consequence of the fact that forces always come in equal and opposite pairs. However, this 'balance' only works if no external force acts on the system from the outside. An external force is like a hand reaching into the pool table and stopping a ball. It adds or removes momentum from the entire system. In many physics problems, we assume these external forces are negligible (too small to matter) during the brief moment of a collision. The true power of this law is that it allows us to predict the outcome of a collision without knowing the details of the force. During a crash, the force between objects is messy. It changes every millisecond as parts bend and compress. Imagine a 2,000 kg train car rolling at 5 m/s hitting a stationary 3,000 kg car. They latch together. Since $10,000 = 5,000 \times Velocity$, we immediately know the coupled trains move at 2 m/s. We never needed to know the force of the latch. Conservation laws are the 'shortcuts' of physics. They tell us that as long as we keep the system closed, the totals are guaranteed. Yes. As long as you add up the momentum vectors of every single piece, the total will still equal the initial momentum of the system before it broke. Technically, no system on Earth is perfectly isolated. However, for very fast interactions like a bat hitting a ball, the external force of friction is so small compared to the impact force that we can treat it as conserved for that moment. No. Internal forces always come in pairs that cancel out. To change the total momentum of a system, a force must come from an object outside that system's boundary. If your system is just 'the car,' momentum is lost. But if your system is 'the car + the Earth,' the momentum is transferred to the Earth. Because the Earth is so massive, its change in velocity is too small to detect. Velocity doesn't account for mass. A ping-pong ball and a bowling ball at the same speed have very different effects in a collision. Momentum ($mv$) combines both to give a true measure of an object's 'impact potential.' Chapter: Elastic Collisions When two hard objects hit each other and bounce away without changing shape, we are witnessing a specific type of interaction called an elastic collision. Think of the sharp 'clack' when two billiard balls strike. They don't stick together, and they don't flatten like a piece of clay would. In physics, a collision is considered elastic only if the objects involved return perfectly to their original state after the impact. Because the objects don't waste energy changing shape or generating heat, they follow two very strict conservation rules. You already know that momentum is always conserved in any closed system where objects collide. However, elastic collisions have a second 'special' rule: the total kinetic energy (the energy of motion) is also conserved. Because energy was used to physically deform the metal. That energy is 'lost' from the motion, meaning kinetic energy was not conserved. Let's look at a classic example: two identical steel balls on a track. In the real world, you might notice that even 'bouncy' things eventually stop. A tennis ball dropped on the floor won't bounce back to its original height. This is because perfectly elastic collisions are mostly found at the atomic level, between gas molecules or subatomic particles. For large objects, some energy always escapes as vibrations (sound) or a tiny bit of heat. So, while most collisions we see are 'mostly' elastic, the perfect version is a useful ideal where no energy is wasted. Energy is always conserved in the *entire universe*, but in a single collision, *kinetic energy* (the energy of motion) can be converted into other forms like heat or sound. Once that motion energy is turned into heat in the floor or sound in the air, the ball no longer has the 'fuel' to keep bouncing. No, but it is very close! A superball can return about 80-90% of its kinetic energy. You can tell it's not perfect because you hear a 'pop' when it hits the ground and it won't bounce back to exactly the same height you dropped it from. No. This is a common point of confusion. Momentum is always conserved in any collision, regardless of whether it's bouncy, sticky, or destructive. Only kinetic energy is picky about the type of collision. When a ball hits the ground and squishes, its molecules are rubbed against each other. This rubbing creates heat, just like rubbing your hands together. That heat takes away from the energy the ball needs to bounce back up. At the atomic level, particles don't have 'surfaces' that can get dented or create friction. They repel each other through fundamental forces (like the electromagnetic force). Since there's no physical squishing of material, no energy is lost to heat or deformation. Chapter: Inelastic Collisions A modern car bumper is designed to deform and crumple when it hits an object during a low-speed collision. Unlike a rubber ball that bounces back to its original shape, the bumper stays bent after the impact. This permanent change in shape is the hallmark of an inelastic collision, where objects do not return to their initial state. In an inelastic collision, some of the initial motion energy is used to rearrange the atoms of the object. This 'work' done to bend the metal or compress the ball means there is less energy available to push the object back up. Lower. Because energy was 'spent' on heat and deformation, the ball has less energy to return to its original height. The most extreme version of this occurs when objects don't bounce at all, but instead stick together. Physicists call this a perfectly inelastic collision, where the two separate objects become a single mass. Even though energy is transformed into heat and sound during this 'sticking' process, momentum is still conserved. This means the total momentum before they hit must equal the total momentum of the combined mass afterward. Imagine a $1\text{ kg}$ block moving at $4\text{ m/s}$ hitting a stationary $1\text{ kg}$ block. The total momentum before is $1 \times 4 = 4 \text{ kg}\cdot\text{m/s}$. After they stick, the new mass is $2\text{ kg}$. To keep the momentum at $4$, the new velocity must be $2\text{ m/s}$. It is important to remember that 'lost' energy does not mean 'lost' momentum. Momentum depends only on mass and velocity, regardless of how hot or deformed the objects become. In the real world, almost every collision you see is at least partially inelastic. Whether it is a football hitting a helmet or a bird hitting a window, energy is always shifting into heat and sound. In an inelastic collision, the crumpling process 'spends' kinetic energy on deforming the metal. This increases the time of the impact and reduces the force transferred to the passengers. An elastic car would bounce violently, causing much higher acceleration and risk of injury. Yes! In fact, most real-world collisions are partially inelastic. This means the objects bounce off each other, but they don't reach their original height or speed because some energy was lost to sound, heat, or minor shape changes. Things stop because of external forces like friction with the ground or air resistance. Momentum is only perfectly conserved in a 'closed system' where these outside forces aren't interfering. In a lab or in space, they would keep moving forever. No. They only stop if their initial momenta were equal and opposite (like two identical cars hitting head-on at the same speed). Otherwise, the combined mass will continue to move in the direction of the object that had more initial momentum. Look for three things: 1. Did the objects stick together? (Perfectly inelastic). 2. Was there a loud sound or heat produced? 3. Did the objects change shape or fail to bounce back to their original speed? If any of these are true, it was an inelastic collision.

From the Knowverse course text.

Chapter: Mechanical Work If you hold a heavy box at shoulder height for ten minutes, your muscles will ache and you will feel exhausted, yet in the world of physics, you have actually done zero work. This happens because the scientific definition of work is much stricter than our everyday use of the word effort. Because the box hasn't moved. In physics, work requires both force and a change in position, known as displacement. For mechanical work to occur, a force must act upon an object and cause that object to move across a specific distance. Consider pushing a stalled car. If you push with all your might but the car stays still, the displacement is zero, so you have done no work on the car. Once the car begins to roll, you are finally performing mechanical work because your force is now causing movement. We measure work in a unit called the Joule (J). One Joule is the work done when a force of one Newton moves an object one meter. However, forces aren't always perfectly aligned with the direction of motion. This changes how we calculate the work done. Imagine you are pulling a child on a sled by a rope. You pull the rope at an upward angle, but the sled moves horizontally across the snow. In the sled example, only the horizontal component of your pull is actually doing work to move the sled forward. The part of your force that pulls straight up is 'wasted' in terms of horizontal movement, though it might make the sled slightly lighter on the snow. If you pull perfectly horizontally, the angle is 0 degrees. Since the cosine of 0 is 1, the formula simplifies back to our basic version. But what happens if the force is applied in a direction completely different from the motion? Consider a waiter carrying a heavy tray of food across a room at a constant speed. The waiter applies an upward force to keep the tray from falling. However, the tray is moving horizontally across the floor. Zero. Because the force is perpendicular (90 degrees) to the motion, none of that force contributes to the horizontal displacement. Mathematically, the cosine of 90 degrees is 0. When you multiply by zero in our formula, the total work becomes zero. This principle also applies to objects moving in a circle, like a satellite orbiting Earth. Gravity pulls the satellite toward the center of the Earth, but the satellite moves along its circular path. At every point, the force is perpendicular to the motion. Therefore, gravity does zero work on a satellite in a perfectly circular orbit, meaning its speed remains constant. To summarize, work isn't just about how hard you try; it's about how much of your force actually helps move the object. The energy is being used by your body to keep your muscles contracted. On a microscopic level, your muscle fibers are constantly sliding and doing work *internally*, but because the box itself doesn't move, no work is done *on the box*. Work is a scalar quantity. Even though it is calculated using two vectors (force and displacement), it only has magnitude and no direction. You don't say '50 Joules North'; you just say '50 Joules'. Yes! Work is negative when the force acts in the opposite direction of the motion (an angle of 180 degrees). A common example is friction, which does negative work to slow down a sliding object. If the force changes while the object moves, you can't use the simple $W = Fd$ formula. Instead, you would calculate the work by finding the area under the curve on a Force-Distance graph. Yes. Because you are moving the backpack upward (displacement) and applying an upward force to lift it, you are doing work against gravity. The vertical portion of your movement counts toward the work done. Chapter: Defining Power If two people lift identical heavy boxes to the same shelf, they both perform exactly the same amount of mechanical work. However, if one person finishes the task in two seconds while the other takes ten, there is a clear difference in the intensity of their effort. In physics, we define Power as the rate at which work is done over a specific period. This means that if you do the same amount of work in less time, you are using more power. Motor A is more powerful because it accomplishes the same amount of work in a shorter amount of time. The standard unit for measuring power is the Watt, named after the inventor James Watt. One Watt is defined as doing one Joule of work every single second. In summary, power tells us the urgency of energy transfer; it bridges the gap between 'how much' work is done and 'how fast' it happens. Not necessarily. Power only tells you how *fast* energy is transferred, not how much of that energy is actually doing useful work versus being wasted as heat. A very powerful engine could still be very inefficient. Because a single Watt is a very small unit. A typical car engine produces tens of thousands of Watts. Using kilowatts (1 kW = 1,000 W) makes the numbers easier to read and compare. No. In physics, power is the rate of *work*. Since the wall doesn't move, the work done is zero. Therefore, even if you are sweating and tired, your physical power output in the physics sense is zero. No. Force is a push or pull. Power is how quickly that force performs work over a distance. You can exert a huge force (like holding a heavy weight) with zero power if nothing moves. In basic mechanics, power is usually treated as a scalar (just a number). However, if an object is doing work *on* a system (like a car braking and regenerating energy), we sometimes describe that as negative power relative to the car's motion. Chapter: Potential Energy A Welsh archer pulls back a heavy longbow string, feeling the tension build in his arms as he holds the arrow steady. This state of readiness is called potential energy. It is energy that an object possesses because of its position or configuration. When the archer releases the string, that stored energy will immediately do work on the arrow, launching it forward. No. Work requires movement (force over distance). However, the work already done to pull the string is now stored as potential energy. While a bow stores energy through tension, most everyday examples of potential energy involve gravity. When you lift a heavy box from the floor to a high shelf, you are doing work against the pull of gravity. This specific type is called gravitational potential energy (GPE). It depends on three specific factors. If you double the mass of the box, you double the energy. If you lift it twice as high, you also double the energy. We can express this relationship with a simple mathematical formula. The 'height' in our formula is actually relative. This means we have to choose where 'zero' is. Imagine a book sitting on a table in a room on the second floor of a building. In most physics problems, we choose the ground or the lowest point in the system as our zero level. Yes. The 'h' in the formula increases because the bottom of the pool is further away, so the calculated potential energy increases. Potential energy isn't just a number; it represents the stored work that is ready to be released. Whether it is a stretched bow or a boulder on a cliff, potential energy is a hidden capacity to make things happen. No. While gravitational potential energy is the most common type discussed, energy can also be stored in other ways, such as in a compressed spring (elastic potential energy) or in chemical bonds (chemical potential energy). On Earth, gravity accelerates all objects at a rate of approximately 9.8 meters per second squared. Since GPE is work done against this specific force, we must include it to get an accurate measure of stored energy. Yes, if the object is below your chosen reference level. For example, if you define the ground as zero and you drop a ball into a 5-meter deep hole, its height becomes -5, resulting in negative potential energy relative to the ground. As the object falls, its height decreases, so its potential energy decreases. That energy isn't lost; it is transformed into kinetic energy (the energy of motion), which you will learn about in the next chapter. No. GPE only depends on the total mass, the strength of gravity, and the height of its center. A 1kg block of lead and a 1kg bag of feathers at the same height have the exact same gravitational potential energy. Chapter: Kinetic Energy A baseball sitting quietly on a patch of grass has no ability to knock anything over on its own. But once a pitcher throws that same ball, it gains a new property: the ability to do work on whatever it hits, like a catcher's mitt or a wooden bat. This energy that an object possesses specifically because it is in motion is called kinetic energy. If an object isn't moving, its kinetic energy is exactly zero, no matter how heavy it is. The amount of kinetic energy an object has depends on two main factors: its mass and its speed. Physicists express this relationship with the formula: $KE = \frac{1}{2}mv^2$. Because speed is squared, small increases in how fast you go lead to massive increases in energy. Consider a car trying to stop. The brakes must remove all the car's kinetic energy to bring it to a halt. Let's pause here. It is a common mistake to think that going twice as fast just means twice the danger. But the square in the formula tells us it's much more significant. Now, where does this energy come from? It comes from the work done on the object. The Work-Energy Theorem states that the net work done on an object is equal to its change in kinetic energy. Since you did twice the work (Force × Distance), the car will gain twice as much kinetic energy. In previous chapters, we saw that Potential Energy is stored. Here, we see that kinetic energy is that storage 'in action.' Work is the bridge that moves energy between these states. The $1/2$ comes from the mathematical derivation using calculus. When you integrate the formula for work ($F \times d$) and substitute acceleration ($a$), the math naturally results in that $1/2$ constant. Without it, the units of energy and work wouldn't match up perfectly. No. Mass is always positive, and because velocity is squared ($v^2$), even a negative velocity (moving backward) becomes positive when squared. Therefore, kinetic energy is always zero or a positive value. No. Kinetic energy is a scalar quantity, meaning it only has magnitude, not direction. A car going 50 mph North has the exact same kinetic energy as the same car going 50 mph East. When you use brakes to stop a car, the kinetic energy doesn't just vanish. According to the Work-Energy Theorem, the 'negative work' done by the brakes turns that motion energy into thermal energy (heat). This is why brake discs get extremely hot after a long downhill drive. The energy would increase by 8 times. Doubling the mass doubles the energy ($ imes 2$). Doubling the speed quadruples the energy ($ imes 4$). Multiplying those together ($2 \times 4$) gives you an 8-fold increase. Chapter: Energy Conservation A heavy pendulum swinging in a room eventually slows down and comes to a complete rest, raising a fundamental question about what happened to its initial energy. When the pendulum is at its highest point, it has maximum potential energy. As it swings down, that energy turns into kinetic energy. In a perfect world with no air, this back-and-forth trade would continue forever without losing a single joule of energy. This leads us to the Law of Conservation of Energy, which states that energy cannot be created or destroyed. Energy can only be transformed from one form to another, meaning the total amount in the universe is always the same. It would swing for a much longer time, but eventually, friction at the pivot point (the hinge) would still convert its energy into heat. To see this law work perfectly, we must look at an isolated system. An isolated system is a physical environment that does not exchange matter or energy with its surroundings. If energy cannot be destroyed, why do machines like cars eventually run out of it? The answer lies in transformation. Energy often moves into forms that are no longer useful to us, like heat. James Joule proved that the 'lost' potential energy of the weights was exactly equal to the new thermal energy in the water. When you rub your hands together, the mechanical work of your muscles is transformed into heat through friction. The energy hasn't disappeared; it has simply moved from the large-scale motion of your hands to the small-scale vibration of molecules. Because no transformation is 100% efficient. A large portion of the chemical energy in fuel is 'lost' as thermal energy during the combustion process. We call this 'lost' energy unusable because once it scatters as heat into the environment, we can't easily collect it to do work again. However, if we could measure every single vibrating molecule in the room, the total energy would still match what we started with. In conclusion, energy is a constant quantity in our universe that simply wears many different masks. Whether it is the height of a pendulum or the warmth of a brake pad, the total balance never changes. While the *total* amount of energy in the universe never changes, the amount of usable energy does. When we use electricity to light a room, that high-quality energy is transformed into low-quality heat that scatters into the environment. We can't 're-collect' that heat to power the light again, so we must constantly generate new usable energy from sources like fuel or wind. In the real world, no. Every mechanical process involves some form of friction or resistance, which inevitably transforms some energy into heat. Even the most advanced electric motors or turbines lose a small percentage of energy to thermal forms, making 100% efficiency an 'ideal' that we can approach but never perfectly reach. A closed system can exchange energy (like heat or light) with its surroundings but cannot exchange matter (like air or water). An isolated system is more restrictive; it can exchange neither energy nor matter. A sealed pot on a stove is closed; a perfectly insulated vacuum flask is an attempt at being isolated. Not necessarily! It depends on the goal. In a car engine, heat is a 'loss' because we want motion. However, in a toaster or an electric heater, heat is the desired output. In those cases, the 'loss' would actually be any energy that turns into light or sound instead of heat. Einstein's famous equation showed that mass itself is a form of 'locked-up' energy. In some high-energy reactions (like in the sun), mass can be transformed into pure energy. This expanded our understanding: the total sum of mass plus energy is what is truly conserved in the universe.

From the Knowverse course text.