Sure! To find the kinetic energy of the arrow at the moment it leaves the bow, we can use the kinetic energy formula:
[tex]\[ KE = \frac{1}{2} m v^2 \][/tex]
where:
- [tex]\( KE \)[/tex] is the kinetic energy,
- [tex]\( m \)[/tex] is the mass of the arrow (in kilograms),
- [tex]\( v \)[/tex] is the velocity of the arrow (in meters per second).
Now, let’s plug in the given values:
- The mass [tex]\( m \)[/tex] of the arrow is [tex]\( 0.155 \)[/tex] kg,
- The velocity [tex]\( v \)[/tex] of the arrow is [tex]\( 31.4 \)[/tex] m/s.
Step-by-step calculation:
1. Square the velocity:
[tex]\[
v^2 = (31.4)^2 = 985.96 \, \text{m}^2/\text{s}^2
\][/tex]
2. Multiply the mass by the squared velocity:
[tex]\[
m \cdot v^2 = 0.155 \times 985.96 = 152.8338 \, \text{kg} \cdot \text{m}^2/\text{s}^2
\][/tex]
3. Multiply by [tex]\( \frac{1}{2} \)[/tex] to find the kinetic energy:
[tex]\[
KE = \frac{1}{2} \cdot 152.8338 = 76.4169 \, \text{J}
\][/tex]
So, the kinetic energy of the arrow the moment it leaves the bow is approximately [tex]\( 76.4169 \)[/tex] Joules.
