An archer fires an arrow, by applying an average force of 182 N across a distance of 38.0 cm on the bow. If the arrow has a mass of 22.0 g, find its velocity when the archer releases the bow:
a)if the conversion is 100% efficient.
b)if the conversion is 77.0% efficient.
work = energy
a) 182 * .380 = 1/2 * .0220 * v^2
b) .770 * 182 * .380 = 1/2 * .0220 * v^2
To calculate the velocity of the arrow when the archer releases the bow, we can use the principle of work and energy.
a) If the conversion is 100% efficient, all the potential energy stored in the bow is converted into kinetic energy of the arrow. We can calculate the potential energy stored in the bow using the formula:
Potential Energy = Force * Distance
In this case, the force applied is 182 N and the distance is 38.0 cm (which we'll convert to meters by dividing by 100):
Potential Energy = 182 N * 0.38 m = 69.16 Joules
Since the potential energy is converted entirely into kinetic energy, we can equate the two using the formula:
Kinetic Energy = (1/2) * Mass * Velocity^2
In this case, the mass of the arrow is 22.0 g (which we'll convert to kilograms by dividing by 1000). Let's assume the velocity of the arrow when released is v.
69.16 Joules = (1/2) * 0.022 kg * v^2
v^2 = (69.16 Joules) / (0.011 kg)
v^2 = 6287.27 m^2/s^2
v = √(6287.27 m^2/s^2)
v ≈ 79.2 m/s
Therefore, the velocity of the arrow when the archer releases the bow, assuming 100% efficiency, is approximately 79.2 m/s.
b) If the conversion is 77.0% efficient, we need to account for the energy losses. In this case, the kinetic energy of the arrow will be only 77.0% of the potential energy stored in the bow.
Potential Energy = 69.16 Joules
Kinetic Energy = 77.0% of Potential Energy = 0.77 * 69.16 Joules = 53.2552 Joules
Using the same formula as above:
53.2552 Joules = (1/2) * 0.022 kg * v^2
v^2 = (53.2552 Joules) / (0.011 kg)
v^2 = 4841.38 m^2/s^2
v = √(4841.38 m^2/s^2)
v ≈ 69.6 m/s
Therefore, the velocity of the arrow when the archer releases the bow, assuming 77.0% efficiency, is approximately 69.6 m/s.