To find the speed of the mass just before it touches the spring, we begin by determining the spring constant,
k, using Hooke's Law, where the force
F is proportional to the compression
x :
F=kxGiven that a force of 100 N compresses the spring by 1 m , the spring constant
k is calculated as:
k=‌=‌=100N∕mNext, we apply the principle of energy conservation. The initial potential and kinetic energy will equal the final potential energy stored in the compressed spring. The relevant energy conservation equation is:
‌mv2+mgh=‌kxmax2where:
m is the mass of the block ( 10 kg ),
g is the acceleration due to gravity
(10m∕ s2),
h is the vertical height change, which is the same as the compression in this scenario (
xmax=2m ),
v is the speed just before touching the spring,
x‌max ‌ is the maximum compression of the spring ( 2 m ).
Rearranging the equation for
v gives:
v=√‌−2ghSubstituting the known values:
v=√‌−2(10m∕ s2)(‌)This simplifies to:
v=√‌−40 v=√40−20=√20m∕ sThus, the speed of the mass just before it touches the spring is
√20m∕ s