Question

A wagon of mass 150 tonnes, starts from rest and travels 50 metres down a
5% grade and strikes a post with bumper spring as shown in Figure below.If the rolling resistance of the track is 50 N/t, find
a)Thevelocity with which the wagon strikes the post.
b)Also find the amount by which the spring will be compressed, if the bumper spring
compresses 1 mm per 25kNforce.

Answer 1

a)

Using F=ma

1471500Sin2.86-50*150=150*1000*a

a=0.439m/s²

Now using v²=u²+2as

v²=0+2*0.439*50

v=6.625m/s

b)

E₁=$\frac{1}{2}mv^{2}+mgh_1$

$E_2=mgh_2$

$E_3=\frac{1}{2}kx^{2}$

$x$ is the compression of spring

$E_4=$ rolling resistance * $x$

$E_1=E_2+E_3+E_4$

$\frac{1}{2}mv^{2}+mgh_1=mgh_2+\frac{1}{2}kx^{2}+50*150*x$

$\frac{1}{2}mv^{2}=mg(h_2-h_1)+\frac{1}{2}kx^{2}+50*150*x$

$\frac{1}{2}mv^{2}=mg*Sin\theta +\frac{1}{2}kx^{2}+50*150*x$

$\frac{1}{2}*150000*(6.625)^{2}=150000*9.81*Sin2.86+0.5$

$3292796.875=73421.50x+\frac{1}{2}*25000000x^{2}+7500x$

$391796.875=73421.5x+12500000x^{2}+7500x$

$12500000xx^{2}+80921.5x-3291796.875=0$

x=0.50m