The maximum speed with which this car can round a second unbanked curve is 40 m/s
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Centripetal Acceleration can be formulated as follows:
[tex]\large {\boxed {a = \frac{ v^2 } { R } }[/tex]
a = Centripetal Acceleration ( m/s² )
v = Tangential Speed of Particle ( m/s )
R = Radius of Circular Motion ( m )
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Centripetal Force can be formulated as follows:
[tex]\large {\boxed {F = m \frac{ v^2 } { R } }[/tex]
F = Centripetal Force ( m/s² )
m = mass of Particle ( kg )
v = Tangential Speed of Particle ( m/s )
R = Radius of Circular Motion ( m )
Let us now tackle the problem !
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Given:
initial speed of the car = v₁ = 20 m/s
radius of first curve = R₁ = 80 m
radius of second curve = R₂ = 320 m
Asked:
final speed of the car = v₂ = ?
Solution:
Firstly , we will derive the formula to calculate the maximum speed of the car:
[tex]\Sigma F = ma[/tex]
[tex]f = m \frac{v^2}{R}[/tex]
[tex]\mu N = m \frac{v^2}{R}[/tex]
[tex]\mu m g = m \frac{v^2}{R}[/tex]
[tex]\mu g = \frac{v^2}{R}[/tex]
[tex]v^2 = \mu g R[/tex]
[tex]\boxed {v = \sqrt { \mu g R } }[/tex]
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Next , we will compare the maximum speed of the car on the first curve and on the second curve:
[tex]v_1 : v_2 = \sqrt { \mu g R_1 } : \sqrt { \mu g R_2 }[/tex]
[tex]v_1 : v_2 = \sqrt { R_1 } : \sqrt { R_2 }[/tex]
[tex]20 : v_2 = \sqrt { 80 } : \sqrt { 320 }[/tex]
[tex]20 : v_2 = 1 : 2[/tex]
[tex]v_2 = 2(20)[/tex]
[tex]\boxed {v_2 = 40 \texttt{ m/s}}[/tex]
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Grade: High School
Subject: Physics
Chapter: Circular Motion
