## Kinetic Energy [block_0] **Kinetic energy **is the energy an object possesses due to its motion. It depends on two factors: 1. **Mass **of the object ($m$) 2. **Speed **of the object ($v$) The formula for **kinetic energy** is: $$E_k = \frac{1}{2}mv^2$$ [{"_key":"block_9","_type":"block","asset":null,"children":[{"_key":"03110c6733bd","_type":"span","markDefs":[],"marks":["strong"],"text":"Kinetic energy "},{"_key":"61427fa80843","_type":"span","markDefs":[],"marks":[],"text":"is a "},{"_key":"d30615052ac8","_type":"span","markDefs":[],"marks":["strong"],"text":"scalar "},{"_key":"4b43a73e843f","_type":"span","markDefs":[],"marks":[],"text":"quantity, meaning it has magnitude but no direction."}],"markDefs":[],"style":"normal"}] A car of mass 1000 kg is moving at a speed of $20 \ \text{m s}^{-1}. What is its kinetic energy? * Use the formula for kinetic energy:$$E_k = \frac{1}{2}mv^2$$ * Substitute the values:$$E_k = \frac{1}{2} \times 1000 \, \text{kg} \times (20 \, \text{m s}^{-1})^2$$ * Calculate:$$E_k = 200,000 \, \text{J}$$ * The car's kinetic energy is 200,000 J (joules). ## Potential Energy [block_15] **Potential energy **is the energy stored in an object due to its position or configuration. There are two main types of potential energy: 1. **Gravitational Potential Energy** 2. **Elastic Potential Energy** ### Gravitational Potential Energy [block_21] **Gravitational potential energy **is the energy stored due to the position of an object in a gravitational field. The formula for **gravitational potential energy** is: $$E_p = mgh$$ where: 1. $m$ is the mass of the object 2. $g$ is the acceleration due to gravity (approximately $9.81 \ \text{m s}^{-2}$ on Earth) 3. $h$ is the height above the reference point A rock of mass 5 kg is lifted to a height of 10 m. What is its gravitational potential energy? * Use the formula for gravitational potential energy:$$E_p = mgh$$ * Substitute the values:$$E_p = 5 \, \text{kg} \times 9.81 \, \text{m s}^{-2} \times 10 \, \text{m}$$ * Calculate:$$E_p = 490.5 \, \text{J}$$ * The rock's gravitational potential energy is 490.5 J. [{"_key":"e924b4acdce5","_type":"block","asset":null,"children":[{"_key":"9fa70fc785400","_type":"span","marks":[],"text":"Note that $E_p = mgh$ is only valid for "},{"_key":"e90e2bd793ff","_type":"span","marks":["strong"],"text":"small height "},{"_key":"5716e8c8502e","_type":"span","marks":[],"text":"changes near a planet's surface. "}],"level":1,"listItem":"bullet","markDefs":[],"style":"normal"},{"_key":"66cb8f0db149","_type":"block","asset":null,"children":[{"_key":"a1fa52e4be32","_type":"span","marks":[],"text":"For "},{"_key":"8db7f0d61447","_type":"span","marks":["strong"],"text":"larger "},{"_key":"2070fcd7c7d4","_type":"span","marks":[],"text":"distances, the formula $E_p = -\\frac{GMm}{r}$ must be used."}],"level":1,"listItem":"bullet","markDefs":[],"style":"normal"}] ### Elastic Potential Energy [block_35] **Elastic potential energy **is the energy stored in an elastic object, such as a spring, when it is compressed or stretched. The formula for **elastic potential energy** is: $$E_e = \frac{1}{2}kx^2$$ where: 1. $k$ is the spring constant (a measure of the spring's stiffness) 2. $x$ is the displacement from the spring's equilibrium position A spring with a spring constant of $200 \ \text{N m}^{-1}$ is compressed by 0.1 m. What is the elastic potential energy stored in the spring? * Use the formula for elastic potential energy:$$E_e = \frac{1}{2}kx^2$$ * Substitute the values:$$E_e = \frac{1}{2} \times 200 \, \text{N m}^{-1} \times (0.1 \, \text{m})^2$$ * Calculate:$$E_e = 1 \, \text{J}$$ * The elastic potential energy stored in the spring is 1 J. ![](https://cdn.sanity.io/images/w7j7fz60/production/14e6b4663e8119d42a7982f14c0779731a6cbf9c-2048x1548.png) ## Mechanical Energy [block_48] **Mechanical energy **is the sum of an object's kinetic energy and potential energy. In an ideal system (without resistive forces like friction or air drag), the **total **mechanical energy remains **constant**. > This is known as the **conservation of mechanical energy**. ### Conservation of Mechanical Energy [block_52] In an isolated system, the total mechanical energy is conserved: $$E_{\text{total}} = E_k + E_p = \text{constant}$$ ![](https://cdn.sanity.io/images/w7j7fz60/production/7b078cb0b3152ffbdc4b6a21fe6805ced28885c5-800x412.png) [{"_key":"28f86f3f7829","_type":"block","asset":null,"children":[{"_key":"25c00cb5d654","_type":"span","marks":[],"text":"Consider the figure below."}],"markDefs":[],"style":"normal"},{"_key":"block_56","_type":"block","asset":null,"children":[{"_key":"613f75e8699a","_type":"span","markDefs":[],"marks":[],"text":"A pendulum swings from its "},{"_key":"cfab305e31a5","_type":"span","markDefs":[],"marks":["strong"],"text":"highest "},{"_key":"88c280e6ab0b","_type":"span","markDefs":[],"marks":[],"text":"point (for example, A) to its "},{"_key":"5dcc360c2731","_type":"span","markDefs":[],"marks":["strong"],"text":"lowest "},{"_key":"7e9846b57c66","_type":"span","markDefs":[],"marks":[],"text":"point (for example, B). "}],"level":1,"listItem":"bullet","markDefs":[],"style":"normal"},{"_key":"c414fe76f558","_type":"block","asset":null,"children":[{"_key":"a07a596b55f1","_type":"span","markDefs":[],"marks":[],"text":"At "},{"_key":"6d529894d189","_type":"span","markDefs":[],"marks":["strong"],"text":"point A"},{"_key":"61a7f92b4c42","_type":"span","markDefs":[],"marks":[],"text":", the pendulum has "},{"_key":"33fe80c450d0","_type":"span","markDefs":[],"marks":["strong"],"text":"maximum "},{"_key":"234d8b9c4889","_type":"span","markDefs":[],"marks":[],"text":"potential energy and "},{"_key":"7a8d05b10460","_type":"span","markDefs":[],"marks":["strong"],"text":"zero "},{"_key":"b54499505471","_type":"span","markDefs":[],"marks":[],"text":"kinetic energy. "}],"level":1,"listItem":"bullet","markDefs":[],"style":"normal"},{"_key":"b1cdd277ac94","_type":"block","asset":null,"children":[{"_key":"0bf12156959a","_type":"span","markDefs":[],"marks":[],"text":"At "},{"_key":"481c331bbb3e","_type":"span","markDefs":[],"marks":["strong"],"text":"point B"},{"_key":"b071bcb6dfb8","_type":"span","markDefs":[],"marks":[],"text":", it has "},{"_key":"e0c165dff50d","_type":"span","markDefs":[],"marks":["strong"],"text":"maximum "},{"_key":"1134916c7116","_type":"span","markDefs":[],"marks":[],"text":"kinetic energy and "},{"_key":"eda416da739d","_type":"span","markDefs":[],"marks":["strong"],"text":"zero "},{"_key":"0768c4660da1","_type":"span","markDefs":[],"marks":[],"text":"potential energy. "}],"level":1,"listItem":"bullet","markDefs":[],"style":"normal"},{"_key":"802bde5395ff","_type":"block","asset":null,"children":[{"_key":"1cea2e720e60","_type":"span","markDefs":[],"marks":[],"text":"The "},{"_key":"d0be496a6481","_type":"span","markDefs":[],"marks":["strong"],"text":"total "},{"_key":"d9c98e6cb78e","_type":"span","markDefs":[],"marks":[],"text":"mechanical energy remains "},{"_key":"b7ac0eccaf28","_type":"span","markDefs":[],"marks":["strong"],"text":"constant "},{"_key":"563b2203cb54","_type":"span","markDefs":[],"marks":[],"text":"throughout the swing."}],"level":1,"listItem":"bullet","markDefs":[],"style":"normal"}] ![](https://cdn.sanity.io/images/w7j7fz60/production/74da18fdf21f66a78cdaee298becb38e0e836395-2048x1645.png) [{"_key":"block_58","_type":"block","asset":null,"children":[{"_key":"684f231f3c29","_type":"span","markDefs":[],"marks":[],"text":"Students often forget that mechanical energy is only conserved"},{"_key":"1665f6dbf450","_type":"span","markDefs":[],"marks":["strong"],"text":" in the absence of resistive forces"},{"_key":"34e2954e923d","_type":"span","markDefs":[],"marks":[],"text":". "}],"level":1,"listItem":"bullet","markDefs":[],"style":"normal"},{"_key":"56fc83e2c550","_type":"block","asset":null,"children":[{"_key":"a6c4422cea21","_type":"span","markDefs":[],"marks":[],"text":"In real-world scenarios, "},{"_key":"57384dad6fac","_type":"span","markDefs":[],"marks":["strong"],"text":"energy losses"},{"_key":"bfd814c66d65","_type":"span","markDefs":[],"marks":[],"text":" due to friction or air drag can "},{"_key":"9375a26350a6","_type":"span","markDefs":[],"marks":["strong"],"text":"reduce "},{"_key":"69ed86c2f638","_type":"span","markDefs":[],"marks":[],"text":"the "},{"_key":"35ac4d1a0fc0","_type":"span","markDefs":[],"marks":["strong"],"text":"total "},{"_key":"908f09bc64f3","_type":"span","markDefs":[],"marks":[],"text":"mechanical energy."}],"level":1,"listItem":"bullet","markDefs":[],"style":"normal"}] ## Energy Losses [block_59] 1. In **real **systems, energy is often **lost **due to **resistive forces** such as friction and air drag. 2. These forces convert **mechanical **energy into **thermal **energy, **reducing **the **total **mechanical energy of the system. ### Friction [block_62] **Friction **is a force that opposes the motion of an object. When an object slides across a surface, friction converts some of its **kinetic **energy into **thermal **energy, causing the object to **slow down**. [{"_key":"block_66","_type":"block","asset":null,"children":[{"_key":"d72ebd472fbe","_type":"span","markDefs":[],"marks":[],"text":"A block sliding down a rough incline loses kinetic energy due to friction. "}],"level":1,"listItem":"bullet","markDefs":[],"style":"normal"},{"_key":"6fd02da85624","_type":"block","asset":null,"children":[{"_key":"83af4125fad5","_type":"span","markDefs":[],"marks":[],"text":"This "},{"_key":"1562d52b9162","_type":"span","markDefs":[],"marks":["strong"],"text":"energy "},{"_key":"f403986efa4d","_type":"span","markDefs":[],"marks":[],"text":"is transformed into "},{"_key":"35fe2d19182e","_type":"span","markDefs":[],"marks":["strong"],"text":"heat"},{"_key":"e874059c1a27","_type":"span","markDefs":[],"marks":[],"text":", warming the block and the surface it slides on."}],"level":1,"listItem":"bullet","markDefs":[],"style":"normal"}] ### Air Drag [block_67] **Air drag **is a resistive force that acts on objects moving through air. 1. It **increases **with the object's **speed **and **surface area**. 2. Like friction, air drag converts **kinetic **energy into **thermal **energy, **reducing **the object's **speed**. [{"_key":"block_72","_type":"block","asset":null,"children":[{"_key":"01ffd9153c78","_type":"span","markDefs":[],"marks":[],"text":"A cyclist moving at high speed experiences "},{"_key":"7da0d9b0627c","_type":"span","markDefs":[],"marks":["strong"],"text":"air drag"},{"_key":"68297cab71ea","_type":"span","markDefs":[],"marks":[],"text":", which reduces their kinetic energy and slows them down. "}],"level":1,"listItem":"bullet","markDefs":[],"style":"normal"},{"_key":"fb4cc4e22345","_type":"block","asset":null,"children":[{"_key":"b3a2353b4760","_type":"span","markDefs":[],"marks":[],"text":"This is why cyclists wear streamlined helmets and clothing to "},{"_key":"1c5a1c067e82","_type":"span","markDefs":[],"marks":["strong"],"text":"minimize air resistance"},{"_key":"9ea5b36e6ad6","_type":"span","markDefs":[],"marks":[],"text":"."}],"level":1,"listItem":"bullet","markDefs":[],"style":"normal"}] [{"_key":"block_74","_type":"block","asset":null,"children":[{"_key":"aeb3aa790063","_type":"span","markDefs":[],"marks":[],"text":"In non-ideal systems, the "},{"_key":"805559c33132","_type":"span","markDefs":[],"marks":["strong"],"text":"total "},{"_key":"aca5143b0189","_type":"span","markDefs":[],"marks":[],"text":"mechanical energy "},{"_key":"dff547183e07","_type":"span","markDefs":[],"marks":["strong"],"text":"decreases "},{"_key":"4a132db7179a","_type":"span","markDefs":[],"marks":[],"text":"due to energy losses. "}],"level":1,"listItem":"bullet","markDefs":[],"style":"normal"},{"_key":"4a52bb8fc412","_type":"block","asset":null,"children":[{"_key":"df3d85791de2","_type":"span","markDefs":[],"marks":[],"text":"However, the "},{"_key":"c322e4d35820","_type":"span","markDefs":[],"marks":["strong"],"text":"law of conservation of energy "},{"_key":"80efc5937917","_type":"span","markDefs":[],"marks":[],"text":"still holds: the \""},{"_key":"de5bab1adc77","_type":"span","markDefs":[],"marks":["strong"],"text":"lost"},{"_key":"f011953d0008","_type":"span","markDefs":[],"marks":[],"text":"\" energy is "},{"_key":"53dd5c6c8eea","_type":"span","markDefs":[],"marks":["strong"],"text":"transformed "},{"_key":"f277413a2cc0","_type":"span","markDefs":[],"marks":[],"text":"into other forms, such as thermal energy."}],"level":1,"listItem":"bullet","markDefs":[],"style":"normal"}] [{"_key":"block_82","_type":"block","asset":null,"children":[{"_key":"a441620314ca","_type":"span","marks":[],"text":""}],"markDefs":[],"style":"normal"},{"_key":"block_83","_type":"block","asset":null,"children":[{"_key":"107f41cc4d78","_type":"span","markDefs":[],"marks":[],"text":"What is the formula for kinetic energy?"}],"level":1,"listItem":"bullet","markDefs":[],"style":"normal"},{"_key":"block_84","_type":"block","asset":null,"children":[{"_key":"4c9a894ab56e","_type":"span","markDefs":[],"marks":[],"text":"How does gravitational potential energy differ from elastic potential energy?"}],"level":1,"listItem":"bullet","markDefs":[],"style":"normal"},{"_key":"block_85","_type":"block","asset":null,"children":[{"_key":"176361a9d5ed","_type":"span","markDefs":[],"marks":[],"text":"Under what conditions is mechanical energy conserved?"}],"level":1,"listItem":"bullet","markDefs":[],"style":"normal"},{"_key":"block_86","_type":"block","asset":null,"children":[{"_key":"8c1e479c2892","_type":"span","markDefs":[],"marks":[],"text":"How do resistive forces like friction and air drag affect the total mechanical energy of a system?"}],"level":1,"listItem":"bullet","markDefs":[],"style":"normal"}] ## [block_75]