A 87 g arrow is fired from a bow whose string exerts an average force of 105 N on the arrow over a distance of 76 cm.
1. What is the speed of the arrow as it leaves the bow?
To find the speed of the arrow as it leaves the bow, we can use the work-energy principle, which states that the work done on an object is equal to the change in its kinetic energy. The work done on the arrow can be calculated as the product of the force applied and the distance over which it is applied.
Work done = Force x Distance
Given that the force is 105 N and the distance is 76 cm (convert to meters by dividing by 100):
Work done = 105 N x (76 cm / 100 m/cm) = 79.8 J
Since the work done on the arrow is equal to the change in its kinetic energy, we can equate it to (1/2)mv^2, where m is the mass of the arrow and v is its velocity. Rearranging the equation, we can solve for v:
(1/2)mv^2 = Work done
Speed of the arrow as it leaves the bow:
v = sqrt((2 x Work done) / m)
Plugging in the values, where the mass of the arrow is 87 g (convert to kg by dividing by 1000):
v = sqrt((2 x 79.8 J) / (87 g / 1000 kg/g))
= sqrt((2 x 79.8 J) / (0.087 kg))
= sqrt(1827.586 kg·m^2/s^2 / 0.087 kg)
= sqrt(21024.138 s^2)
≈ 145 m/s
The speed of the arrow as it leaves the bow is approximately 145 m/s.
To find the speed of the arrow as it leaves the bow, we can use the work-energy principle. According to the principle, the work done on an object is equal to its change in kinetic energy.
1. First, let's find the work done on the arrow. The work done can be calculated using the formula: Work = Force * Distance * cos(theta), where theta is the angle between the direction of force and displacement. In this case, the arrow is fired in a straight line, so the angle between the force and displacement is 0 degrees. Therefore, cos(theta) = 1.
Work = Force * Distance * cos(theta) = 105 N * 0.76 m * 1 = 79.8 J
2. Since the work done on the arrow equals its change in kinetic energy, we can equate it to (1/2) * mass * velocity^2.
79.8 J = (1/2) * 0.087 kg * velocity^2
3. Rearranging the equation, we can solve for velocity:
velocity^2 = (2 * 79.8 J) / 0.087 kg
velocity^2 = 1830.34 m^2/s^2
4. Taking the square root of both sides, we find:
velocity ≈ 42.8 m/s
Therefore, the speed of the arrow as it leaves the bow is approximately 42.8 m/s.
force*time= mass*velocity
but time = distance/avgvelocity= distance/finalvelocity/2=2*distance/Vf
2*force*distance= mass*Vf^2
Vf^2=2*force/mass * d