A banked curve means a car should do the turn at the design speed without participation of acceleration. This information allows us to find out the angle of incline of the road:
"\\alpha = \\arctan\\frac{v_0^2}{gR}; \\\\"
For smaller speeds (e.g. car sliding downwards):
"F_n\\sin\\alpha - F_f\\cos\\alpha = m\\frac{v^2}{R}; \\\\\nF_n\\cos\\alpha + F_f\\sin\\alpha - mg = 0; \\\\"
Remembering that "F_f = {\\mu}F_n; \\\\" and dividing the equations we get
"v_{min} = \\sqrt\\frac{gR(\\tan{(\\alpha)}-\\mu)}{1+{\\mu}\\tan\\alpha} = \\sqrt\\frac{gR(v_0^2-{\\mu}gR)}{gR+{\\mu}v_0^2} \\approx 41\\frac{km}{h}. \\\\"
Similarly, for greater speeds (car sliding upwards):
"F_n\\sin\\alpha + F_f\\cos\\alpha = m\\frac{v^2}{R}; \\\\\nF_n\\cos\\alpha - F_f\\sin\\alpha - mg = 0; \\\\"
"v_{max} = \\sqrt\\frac{gR(\\tan{(\\alpha)}+\\mu)}{1-{\\mu}\\tan\\alpha} = \\sqrt\\frac{gR(v_0^2+{\\mu}gR)}{gR-{\\mu}v_0^2} \\approx 108\\frac{km}{h}. \\\\"
When doing the calculation please note that km/h should be converted to m/s and back likewise.
Answer: 41, 108.
Comments
nice
Leave a comment