Answer:
a) U = 0 J
k = 0.393 J
E = 0.393 J
b) U = 0.0229J
k = 0.370 J
E = 0.393 J
c) U = 0.0914 J
k = 0.302 J
E = 0.393 J
d) U = 0.206 J
k = 0.187 J
E = 0.393 J
e) U = 0.366 J
k = 0.027 J
E = 0.393 J
Explanation:
Hi there!
The equations of kinetic energy and elastic potential energy are as follows:
k = 1/2 · m · v²
U = 1/2 · ks · x²
Where:
m = mass of the block.
v = velocity.
ks = spring constant.
x = displacement of the string.
a) When the spring is not compressed, the spring potential energy will be zero:
U = 1/2 · ks · x²
U = 1/2 · 457 N/m · (0 cm)²
U = 0 J
The kinetic energy of the block will be:
k = 1/2 · m · v²
k = 1/2 · 1.05 kg · (0.865 m/s)²
k = 0.393 J
The mechanical energy will be:
E = k + U = 0.393 J + 0 J = 0.393 J
This energy will be conserved, i.e., it will remain constant because there is no work done by friction nor by any other dissipative force (like air resistance). This means that the kinetic energy will be converted only into spring potential energy (there is no thermal energy due to friction, for example).
b) The spring potential energy will be:
U = 1/2 · 457 N/m · (0.01 m)²
U = 0.0229 J
Since the mechanical energy has to remain constant, we can use the equation of mechanical energy to obtain the kinetic energy:
E = k + U
0.393 J = k + 0.0229 J
0.393 J - 0.0229 J = k
k = 0.370 J
c) The procedure is now the same. Let´s calculate the spring potential energy with x = 0.02 m.
U = 1/2 · 457 N/m · (0.02 m)²
U = 0.0914 J
Using the equation of mechanical energy:
E = k + U
0.393 J = k + 0.0914 J
k = 0.393 J - 0.0914 J = 0.302 J
d) U = 1/2 · 457 N/m · (0.03 m)²
U = 0.206 J
E = 0.393 J
k = E - U = 0.393 J - 0.206 J
k = 0.187 J
e) U = 1/2 · 457 N/m · (0.04 m)²
U = 0.366 J
E = 0.393 J
k = E - U = 0.393 J - 0.366 J = 0.027 J.
and r=40 m, so we can re-arrange the previous equation to find the velocity of the car: