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 85 cm. What is the speed of the arrow as it leaves the bow?
work= KE
95*.85=1/2 .085*v^2
To find the speed of the arrow as it leaves the bow, we can use the principle of work and energy. The work done by the bowstring on the arrow will be equal to the change in kinetic energy of the arrow.
Step 1: Convert the given distance from centimeters to meters.
85 cm = 0.85 m
Step 2: Calculate the work done by the bowstring.
Work = Force × Distance
Work = 95 N × 0.85 m
Work = 80.75 J (Joules)
Step 3: Calculate the change in kinetic energy of the arrow.
Change in kinetic energy = Work
Change in kinetic energy = 80.75 J (Joules)
Step 4: Use the formula for kinetic energy to find the speed of the arrow.
Kinetic energy = 1/2 × mass × speed^2
80.75 J = 1/2 × 0.085 kg × speed^2
Step 5: Solve for the speed of the arrow.
speed^2 = (2 × 80.75 J) / (0.085 kg)
speed^2 = 1910 J / kg
speed^2 = 1910
Taking the square root of both sides, we get:
speed = √1910 = 43.7 m/s
Therefore, the speed of the arrow as it leaves the bow is approximately 43.7 m/s.
To find the speed of the arrow as it leaves the bow, we can use the principle of work and energy. The work done on an object is equal to the product of the force applied and the displacement of the object in the direction of the force. In this case, the force applied by the bowstring and the distance over which it is applied are given.
First, we need to find the work done on the arrow. The formula for work is:
Work = Force × Distance × cos(θ)
Where θ is the angle between the force and the direction of the displacement. In this case, the force and displacement are in the same direction, so the angle θ is 0 degrees and the cosine of 0 degrees is 1. So, the formula simplifies to:
Work = Force × Distance
Substituting the given values, we have:
Work = 95 N × 0.85 m = 80.75 J
Next, 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. In this case, the work done on the arrow is equal to the change in its kinetic energy. The kinetic energy of an object is given by the formula:
Kinetic Energy = 1/2 × Mass × Velocity^2
Since the initial velocity of the arrow is 0 (as it starts from rest when released from the bow), the change in kinetic energy is equal to its final kinetic energy. Therefore, we can set up the equation:
Work = 1/2 × Mass × (Velocity Final^2 - Velocity Initial^2)
Substituting the values we already know, we get:
80.75 J = 1/2 × 0.085 kg × (Velocity Final^2 - 0)
Simplifying the equation:
80.75 J = 0.00425 kg × Velocity Final^2
Finally, we can solve for the velocity final by rearranging the equation:
Velocity Final^2 = 80.75 J / 0.00425 kg
Velocity Final^2 = 19000
Taking the square root of both sides:
Velocity Final = √19000
Velocity Final ≈ 137.7 m/s
Therefore, the speed of the arrow as it leaves the bow is approximately 137.7 m/s.