Verified answer

Answer:

The minimum safe speed is 21.44 m/s.

The maximum safe speed is 70.88 m/s.

Explanation:

Given:

• Angle of inclination ( θ ) = 37°
• Coefficient of friction [tex]\sf{\mu}[/tex] = 0.5
• Radius of curvature ( r ) = 250 m
• Acceleration due to gravity ( g ) = 10 m/s²

For minimum safe speed,

[tex]\displaystyle{\boxed{\pink{\bf\:v_{min}\:=\:\sqrt{\dfrac{rg\:(\tan\:\theta\:-\:\mu\:)}{1\:+\:\mu\:\tan\:\theta}\:}}}}[/tex]

[tex]\displaystyle{\implies\sf\:v_{min}\:=\:\sqrt{\dfrac{250\:\times\:10\:(\:\tan\:(\:37^{\circ})\:-\:0.5\:)}{1\:+\:0.5\:\times\:\tan\:(\:37^{\circ}\:)}}}[/tex]

[tex]\displaystyle{\implies\sf\:v_{min}\:=\:\sqrt{\dfrac{2500\:(\:0.753\:-\:0.5\:)}{1\:+\:0.5\:\times\:0.753}}}[/tex]

[tex]\displaystyle{\implies\sf\:v_{min}\:=\:\sqrt{\dfrac{2500\:\times\:0.253}{1\:+\:0.3765}}}[/tex]

[tex]\displaystyle{\implies\sf\:v_{min}\:=\:\sqrt{\dfrac{632.5}{1.3765}}}[/tex]

[tex]\displaystyle{\implies\sf\:v_{min}\:=\:\sqrt{459.498}}[/tex]

[tex]\displaystyle{\implies\sf\:v_{min}\:=\:21.435}[/tex]

[tex]\displaystyle{\therefore\:\underline{\boxed{\red{\sf\:v_{min}\:\approx\:21.44\:m/s\:}}}}[/tex]

For maximum safe speed,

[tex]\displaystyle{\boxed{\blue{\bf\:v_{max}\:=\:\sqrt{\dfrac{rg\:(\tan\:\theta\:+\:\mu\:)}{1\:-\:\mu\:\tan\:\theta}\:}}}}[/tex]

[tex]\displaystyle{\implies\sf\:v_{max}\:=\:\sqrt{\dfrac{250\:\times\:10\:(\:\tan\:(\:37^{\circ})\:+\:0.5\:)}{1\:-\:0.5\:\times\:\tan\:(\:37^{\circ}\:)}}}[/tex]

[tex]\displaystyle{\implies\sf\:v_{max}\:=\:\sqrt{\dfrac{2500\:(\:0.753\:+\:0.5\:)}{1\:-\:0.5\:\times\:0.753}}}[/tex]

[tex]\displaystyle{\implies\sf\:v_{max}\:=\:\sqrt{\dfrac{2500\:\times\:1.253}{1\:-\:0.3765}}}[/tex]

[tex]\displaystyle{\implies\sf\:v_{max}\:=\:\sqrt{\dfrac{3132.5}{0.6235}}}[/tex]

[tex]\displaystyle{\implies\sf\:v_{max}\:=\:\sqrt{5024.057}}[/tex]

[tex]\displaystyle{\therefore\:\underline{\boxed{\purple{\sf\:v_{max}\:=\:70.88\:m/s\:}}}}[/tex]