Explain Kinetic Energy Changes - Middle School Physical Science
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Identify the proportional relationship between $KE$ and mass when speed is constant.
Identify the proportional relationship between $KE$ and mass when speed is constant.
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$KE \propto m$. Direct proportionality means $KE$ increases linearly with mass.
$KE \propto m$. Direct proportionality means $KE$ increases linearly with mass.
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A graph shows $KE$ changes from $25,J$ to $100,J$ at constant mass. What happened to speed?
A graph shows $KE$ changes from $25,J$ to $100,J$ at constant mass. What happened to speed?
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Speed doubled. $\frac{100}{25} = 4 = 2^2$, so speed increased by factor of $2$.
Speed doubled. $\frac{100}{25} = 4 = 2^2$, so speed increased by factor of $2$.
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A graph shows $KE$ changes from $20,J$ to $60,J$ at constant speed. What happened to mass?
A graph shows $KE$ changes from $20,J$ to $60,J$ at constant speed. What happened to mass?
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Mass tripled. $\frac{60}{20} = 3$, so mass increased by factor of $3$.
Mass tripled. $\frac{60}{20} = 3$, so mass increased by factor of $3$.
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Identify the proportional relationship between $KE$ and speed when mass is constant.
Identify the proportional relationship between $KE$ and speed when mass is constant.
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$KE \propto v^2$. Quadratic relationship means $KE$ increases with speed squared.
$KE \propto v^2$. Quadratic relationship means $KE$ increases with speed squared.
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Which variable is squared in $KE = \frac{1}{2}mv^2$: mass or speed?
Which variable is squared in $KE = \frac{1}{2}mv^2$: mass or speed?
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Speed is squared: $v^2$. The exponent on $v$ is $2$, while mass has exponent $1$.
Speed is squared: $v^2$. The exponent on $v$ is $2$, while mass has exponent $1$.
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Identify the graph shape expected for $KE$ vs. speed when mass is constant.
Identify the graph shape expected for $KE$ vs. speed when mass is constant.
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An upward-curving parabola through the origin. Quadratic relationship between $KE$ and speed creates a parabola.
An upward-curving parabola through the origin. Quadratic relationship between $KE$ and speed creates a parabola.
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Identify the graph shape expected for $KE$ vs. mass when speed is constant.
Identify the graph shape expected for $KE$ vs. mass when speed is constant.
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A straight line through the origin. Direct proportionality between $KE$ and mass creates a linear graph.
A straight line through the origin. Direct proportionality between $KE$ and mass creates a linear graph.
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What happens to kinetic energy if speed is cut in half while mass stays the same?
What happens to kinetic energy if speed is cut in half while mass stays the same?
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Kinetic energy becomes $\frac{1}{4}$ as large. Since $KE \propto v^2$, halving speed gives $(\frac{1}{2})^2 = \frac{1}{4}$ the energy.
Kinetic energy becomes $\frac{1}{4}$ as large. Since $KE \propto v^2$, halving speed gives $(\frac{1}{2})^2 = \frac{1}{4}$ the energy.
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What happens to kinetic energy if speed triples while mass stays the same?
What happens to kinetic energy if speed triples while mass stays the same?
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Kinetic energy becomes $9$ times larger. Since $KE \propto v^2$, tripling speed gives $3^2 = 9$ times the energy.
Kinetic energy becomes $9$ times larger. Since $KE \propto v^2$, tripling speed gives $3^2 = 9$ times the energy.
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What happens to kinetic energy if speed doubles while mass stays the same?
What happens to kinetic energy if speed doubles while mass stays the same?
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Kinetic energy becomes $4$ times larger. Since $KE \propto v^2$, doubling speed quadruples kinetic energy.
Kinetic energy becomes $4$ times larger. Since $KE \propto v^2$, doubling speed quadruples kinetic energy.
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What happens to kinetic energy if mass doubles while speed stays the same?
What happens to kinetic energy if mass doubles while speed stays the same?
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Kinetic energy doubles. Since $KE \propto m$, doubling mass doubles kinetic energy.
Kinetic energy doubles. Since $KE \propto m$, doubling mass doubles kinetic energy.
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State the formula for kinetic energy in terms of mass and speed.
State the formula for kinetic energy in terms of mass and speed.
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$KE = \frac{1}{2}mv^2$. Shows kinetic energy depends on mass and speed squared.
$KE = \frac{1}{2}mv^2$. Shows kinetic energy depends on mass and speed squared.
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A $KE$ vs. speed graph shows speed increases by a factor of $3$. By what factor should $KE$ increase?
A $KE$ vs. speed graph shows speed increases by a factor of $3$. By what factor should $KE$ increase?
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$9$. Since $KE \propto v^2$, factor of $3$ in speed gives $3^2 = 9$ in $KE$.
$9$. Since $KE \propto v^2$, factor of $3$ in speed gives $3^2 = 9$ in $KE$.
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If two points on a $KE$ vs. speed graph are $(2, 20)$ and $(4, 80)$, what relationship is shown?
If two points on a $KE$ vs. speed graph are $(2, 20)$ and $(4, 80)$, what relationship is shown?
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Quadratic (doubling speed makes $KE$ $4$ times). $\frac{80}{20} = 4$ and $\frac{4}{2} = 2$, so $KE$ increases as speed squared.
Quadratic (doubling speed makes $KE$ $4$ times). $\frac{80}{20} = 4$ and $\frac{4}{2} = 2$, so $KE$ increases as speed squared.
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If two points on a $KE$ vs. mass graph are $(2, 30)$ and $(4, 60)$, what is the relationship shown?
If two points on a $KE$ vs. mass graph are $(2, 30)$ and $(4, 60)$, what is the relationship shown?
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Direct proportion (doubling mass doubles $KE$). $\frac{60}{30} = 2$ and $\frac{4}{2} = 2$, confirming linear relationship.
Direct proportion (doubling mass doubles $KE$). $\frac{60}{30} = 2$ and $\frac{4}{2} = 2$, confirming linear relationship.
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A graph shows $KE$ changes from $36,J$ to $4,J$ at constant mass. What happened to speed?
A graph shows $KE$ changes from $36,J$ to $4,J$ at constant mass. What happened to speed?
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Speed became $\frac{1}{3}$ as large. $\frac{4}{36} = \frac{1}{9} = (\frac{1}{3})^2$, so speed decreased by factor of $3$.
Speed became $\frac{1}{3}$ as large. $\frac{4}{36} = \frac{1}{9} = (\frac{1}{3})^2$, so speed decreased by factor of $3$.
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A $KE$ vs. mass graph shows mass decreases by a factor of $5$. By what factor should $KE$ change?
A $KE$ vs. mass graph shows mass decreases by a factor of $5$. By what factor should $KE$ change?
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$\frac{1}{5}$. Since $KE \propto m$, factor of $\frac{1}{5}$ in mass gives same factor in $KE$.
$\frac{1}{5}$. Since $KE \propto m$, factor of $\frac{1}{5}$ in mass gives same factor in $KE$.
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A graph compares two objects at the same speed; Object A has twice the mass of B. Which has greater $KE$ and by how much?
A graph compares two objects at the same speed; Object A has twice the mass of B. Which has greater $KE$ and by how much?
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Object A; $2$ times the kinetic energy. At same speed, $KE \propto m$, so twice the mass gives twice the $KE$.
Object A; $2$ times the kinetic energy. At same speed, $KE \propto m$, so twice the mass gives twice the $KE$.
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Which change increases kinetic energy more: doubling mass or doubling speed (starting from same $m$ and $v$)?
Which change increases kinetic energy more: doubling mass or doubling speed (starting from same $m$ and $v$)?
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Doubling speed (it makes $KE$ $4$ times). Doubling speed quadruples $KE$; doubling mass only doubles $KE$.
Doubling speed (it makes $KE$ $4$ times). Doubling speed quadruples $KE$; doubling mass only doubles $KE$.
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Identify the relationship between kinetic energy and speed when mass is constant.
Identify the relationship between kinetic energy and speed when mass is constant.
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Quadratic: $KE \propto v^2$. From $KE = \frac{1}{2}mv^2$, kinetic energy varies with speed squared.
Quadratic: $KE \propto v^2$. From $KE = \frac{1}{2}mv^2$, kinetic energy varies with speed squared.
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Identify the relationship between kinetic energy and mass when speed is constant.
Identify the relationship between kinetic energy and mass when speed is constant.
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Directly proportional: $KE \propto m$. From $KE = \frac{1}{2}mv^2$, kinetic energy varies linearly with mass.
Directly proportional: $KE \propto m$. From $KE = \frac{1}{2}mv^2$, kinetic energy varies linearly with mass.
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What happens to kinetic energy if speed doubles while mass stays constant?
What happens to kinetic energy if speed doubles while mass stays constant?
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Kinetic energy becomes $4$ times larger. Since $KE \propto v^2$, doubling speed quadruples kinetic energy.
Kinetic energy becomes $4$ times larger. Since $KE \propto v^2$, doubling speed quadruples kinetic energy.
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What happens to kinetic energy if mass doubles while speed stays constant?
What happens to kinetic energy if mass doubles while speed stays constant?
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Kinetic energy doubles. Since $KE \propto m$, doubling mass doubles kinetic energy.
Kinetic energy doubles. Since $KE \propto m$, doubling mass doubles kinetic energy.
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State the formula for kinetic energy using mass $m$ and speed $v$.
State the formula for kinetic energy using mass $m$ and speed $v$.
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$KE = \frac{1}{2}mv^2$. Kinetic energy equals half the product of mass and velocity squared.
$KE = \frac{1}{2}mv^2$. Kinetic energy equals half the product of mass and velocity squared.
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On a $KE$ vs $m$ graph at constant $v$, how does the line compare for a larger constant speed?
On a $KE$ vs $m$ graph at constant $v$, how does the line compare for a larger constant speed?
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Steeper slope because slope $= \frac{1}{2}v^2$. Higher speed increases the slope proportionally to $v^2$.
Steeper slope because slope $= \frac{1}{2}v^2$. Higher speed increases the slope proportionally to $v^2$.
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