For the model d = v i 2 /2g( K cos + sin ) , choose a reasonable initial speed,
ID: 1783440 • Letter: F
Question
For the model d = vi2/2g(K cos + sin ) , choose a reasonable initial speed, and a frictional coefficient between 0.40 and 0.90. Given your choices, calculate how far the case would slide when = 0° and 90°. One of these angles gives the longest possible d, but it turns out that the other angle doesn't give the shortest d.
Use a graphing calculator (or trial and error) to find the angle for which the case stops most quickly, and also that minimum distance. Show calculations for 0° and 90°, but the minimum d and the angle that yields it can simply be reported.
Explanation / Answer
Take Vi = 20 m/s and mu(k) = 0.75,
For theta = 0 deg
d = 202 / 2*9.8(0.75*1 + 0) = 27.2 m
for theta = 90
d = 202 / 2*9.8 (0 +1) = 20.4
For theta = 0 , d is maximum.
To find minimum d and corresponding angle we consider angle at interval of 10 deg starting from 80 deg and work till, value of ( muk cos theta + sin theta ) starts decreasing
theta = 85 , 0.75 cos 85 + sin 85 = 1.06
theta = 80 , 0.75 cos 80 + sin 80 = 1.11
theta = 75 , 0.75 cos75 + sin75 = 1.16
theta = 70 , 0.75 cos70 + sin70 = 1.20
theta = 65 , 0.75 cos 65 + sin 65 = 1.22
theta = 60 , 0.75 cos 60 + sin 60 = 1.24
theta = 55 , 0.75 cos 55 + sin 55 = 1.249
theta = 50 , 0.75 cos 50 + sin 50 = 1.248
As value of muk cos theta + sin theta starts decreasing between 55 and 50, minimum d will be for some angle in this range. Exact angle = 53.13 deg
and corresponding value minimum value of d = 16.3 m