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Theory Let's apply energy ideas to the situation of a block that is released from rest at the top of an incline. To start with, we'll assume that the block slides down the ramp without friction. Our goal will be to graph energy as a function of distance traveled, and energy as a function of time. Before doing that, let's think about what we can learn from equations we already know. Constant-acceleration equations (in the special case that the initial velocity is zero): Distance travelled: d=af Speed: V = at and: V² - 2 ad Energy equations: Kinetic energy: K-m Potential energy: U = mgh Work - kinetic energy theorem (special case of no initial velocity): m = Fd Make use of these equations to sketch graphs of energy vs. distance traveled, and energy vs. time, for a block sliding (from rest) down a frictionless incline. Define the zero for gravitational potential energy to be the bottom of the ramp. Plot potential energy, kinetic energy, and total mechanical energy (U+K). energy as a function of distance energy as a function of time Questions If the block slid for twice the distance down a frictionless ramp of the same angle, the kinetic energy at the bottom would, compared to the original, be increased by a factor of √ []2 []4

Theory Let's apply energy ideas to the situation of a block that is released from rest at the top of an incline. To start with, we'll assume that the block slides down the ramp without friction. Our goal will be to graph energy as a function of distance traveled, and energy as a function of time. Before doing that, let's think about what we can learn from equations we already know. Constant-acceleration equations (in the special case that the initial velocity is zero): Distance travelled: d=af Speed: V = at and: V² - 2 ad Energy equations: Kinetic energy: K-m Potential energy: U = mgh Work - kinetic energy theorem (special case of no initial velocity): m = Fd Make use of these equations to sketch graphs of energy vs. distance traveled, and energy vs. time, for a block sliding (from rest) down a frictionless incline. Define the zero for gravitational potential energy to be the bottom of the ramp. Plot potential energy, kinetic energy, and total mechanical energy (U+K). energy as a function of distance energy as a function of time Questions If the block slid for twice the distance down a frictionless ramp of the same angle, the kinetic energy at the bottom would, compared to the original, be increased by a factor of √ []2 []4

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### Theory Let's apply energy ideas to the situation of a block that is released from rest at the top of an incline. To start with, we’ll assume that the block slides down the ramp without friction. Our goal is to graph energy as a function of distance traveled, and energy as a function of time. Before doing that, let’s think about what we can learn from equations we already know. #### Constant-acceleration equations (in the special case that the initial velocity is zero): - **Distance traveled:** \( d = \frac{1}{2} a t^2 \) - **Speed:** \( V = a t \) - **And:** \( V^2 = 2 a d \) #### Energy equations: - **Kinetic energy:** \( K = \frac{1}{2} m V^2 \) - **Potential energy:** \( U = m g h \) - **Work – kinetic energy theorem (special case of no initial velocity):** \[ \frac{1}{2} m V^2 = F_{\text{net}} d \] Make use of these equations to sketch graphs of energy vs. distance traveled, and energy vs. time, for a block sliding (from rest) down a frictionless incline. Define the zero for gravitational potential energy to be the bottom of the ramp. Plot potential energy, kinetic energy, and total mechanical energy (U + K). #### Graphs: - **Energy as a function of distance:** \[ \begin{tikzpicture} \begin{axis}[ xlabel={$d$}, ylabel={energy}, ymin=0, xmin=0, height=6cm, width=\textwidth ] \end{axis} \end{tikzpicture} \] - **Energy as a function of time:** \[ \begin{tikzpicture} \begin{axis}[ xlabel={$t$}, ylabel={energy}, ymin=0, xmin=0, height=6cm, width=\textwidth ] \end{axis} \end{tikzpicture} \] ### Questions If the block slid for twice the distance down a frictionless ramp of the same angle, the kinetic energy at the bottom would, compared to the original, be increased by a factor of:
Transcribed Image Text:### Theory Let's apply energy ideas to the situation of a block that is released from rest at the top of an incline. To start with, we’ll assume that the block slides down the ramp without friction. Our goal is to graph energy as a function of distance traveled, and energy as a function of time. Before doing that, let’s think about what we can learn from equations we already know. #### Constant-acceleration equations (in the special case that the initial velocity is zero): - **Distance traveled:** \( d = \frac{1}{2} a t^2 \) - **Speed:** \( V = a t \) - **And:** \( V^2 = 2 a d \) #### Energy equations: - **Kinetic energy:** \( K = \frac{1}{2} m V^2 \) - **Potential energy:** \( U = m g h \) - **Work – kinetic energy theorem (special case of no initial velocity):** \[ \frac{1}{2} m V^2 = F_{\text{net}} d \] Make use of these equations to sketch graphs of energy vs. distance traveled, and energy vs. time, for a block sliding (from rest) down a frictionless incline. Define the zero for gravitational potential energy to be the bottom of the ramp. Plot potential energy, kinetic energy, and total mechanical energy (U + K). #### Graphs: - **Energy as a function of distance:** \[ \begin{tikzpicture} \begin{axis}[ xlabel={$d$}, ylabel={energy}, ymin=0, xmin=0, height=6cm, width=\textwidth ] \end{axis} \end{tikzpicture} \] - **Energy as a function of time:** \[ \begin{tikzpicture} \begin{axis}[ xlabel={$t$}, ylabel={energy}, ymin=0, xmin=0, height=6cm, width=\textwidth ] \end{axis} \end{tikzpicture} \] ### Questions If the block slid for twice the distance down a frictionless ramp of the same angle, the kinetic energy at the bottom would, compared to the original, be increased by a factor of:
**Question:** If, instead, the block slid for twice the time down a frictionless ramp of the same angle, the kinetic energy at the bottom would, compared to the original, be increased by a factor of: - [ ] √2 - [ ] 2 - [ ] 4
Transcribed Image Text:**Question:** If, instead, the block slid for twice the time down a frictionless ramp of the same angle, the kinetic energy at the bottom would, compared to the original, be increased by a factor of: - [ ] √2 - [ ] 2 - [ ] 4
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