The drawing shows two boxes resting on frictionless ramps. One box is relatively

Question

The drawing shows two boxes resting on frictionless ramps. One box is relatively light and sits on a steep ramp. The other box is heavier and rests on a ramp that is less steep. The boxes are released from rest at A and allowed to slide down the ramps. The two boxes have masses of 8 and 31 kg. If A and B are 3.5 and 1.5 m, respectively, above the ground, determine the speed of (a) the lighter box and (b) the heavier box when each reaches B. (c) What is the ratio of the kinetic energy of the heavier box to that of the lighter box at B?

Explanation / Answer

rom the conservation energy law : reduction of potential energy equals to increase of kinetic energy

for a small box
Ep=m*g*h - reduction of the potential energy
g=10H/kg
h=(A-B)=5m
Ep=13*10*5=650J
Ek=m*V^2/2 - increase of the kinetic energy
so Ep=Ek
m*g*h=m*V^2/2
g*h=V^2/2
V^2=2*g*h
V=sqrt(2*g*h)=sqrt(2*10*5)=sqrt(100)=10

V=10m/s - velocity does NOT depend on mass

so
a) velocity of the lighter box V_light=10m/s
b) velocity of the heavier box V_heavy=10m/s
c) ratio of the kinetic energy
m_heavy=40kg
m_light=13kg
R=Ek_heavy/Ek_light=(m_heavy*V^2/2) / (m_light*V^2/2)=m_heavy / m_light = 40/13= 3.07
R=3.07

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