An 85 g arrow is fired from a bow whose string exerts an average force of 95 N on the arrow over a distance of 81 cm. 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.

First, let's calculate the work done on the arrow by the bow's string. We can use the formula:

Work = Force * Distance * cos(theta)

Where:
- Work is the work done on the arrow (in Joules)
- Force is the average force exerted by the string (in Newtons)
- Distance is the distance over which the force is applied (in meters)
- cos(theta) is the angle between the force and the direction of motion (we'll assume it to be 0 degrees, so cos(theta) = 1)

Plugging in the given values:
Force = 95 N
Distance = 81 cm = 0.81 m (converted to meters)

Work = 95 N * 0.81 m * cos(0°) = 77.295 Joules

Next, we'll use the work-energy principle to find the change in kinetic energy of the arrow. The work done on the arrow will equal the change in its kinetic energy:

Work = Change in Kinetic Energy

Since the arrow starts from rest, its initial kinetic energy is 0.

Change in Kinetic Energy = Final Kinetic Energy - Initial Kinetic Energy

Since the arrow is now moving, its final kinetic energy represents its total kinetic energy at that instant:

Change in Kinetic Energy = 1/2 * mass * velocity^2

Where:
- mass is the mass of the arrow (in kg)
- velocity is the speed of the arrow (in m/s)

Plugging in the given values:
mass = 85 g = 0.085 kg (converted to kg)

77.295 Joules = 1/2 * 0.085 kg * velocity^2

Simplifying the equation:

77.295 Joules = 0.0425 kg * velocity^2

To find the velocity, we can rearrange the equation:

velocity^2 = 77.295 Joules / 0.0425 kg

velocity^2 = 1817.65 m^2/s^2

Taking the square root of both sides:

velocity = √(1817.65 m^2/s^2)

Finally, calculating the result:

velocity ≈ 42.65 m/s

Therefore, the speed of the arrow as it leaves the bow is approximately 42.65 m/s.