An archer's bow applies a force to an arrow over a distance of 19cm, while firing it vertically upwards. If the arrow achieves a speed of 8.0m/s and the efficiency of the bow is 75%, determine the average force applied to the arrow by the bow.
So far I have the Wnet for force input efficiency which is 42.10526316J
To determine the average force applied to the arrow by the bow, we can use the work-energy principle. The work done by the bow is equal to the change in kinetic energy of the arrow.
First, let's calculate the gravitational potential energy gained by the arrow. The arrow is fired vertically upward, so at the highest point of its trajectory, it has maximum potential energy. The potential energy can be calculated using the formula:
Potential Energy (PE) = m * g * h
Where:
m = mass of the arrow
g = acceleration due to gravity
h = maximum height reached by the arrow
Since the arrow is fired vertically upward and eventually stops at its highest point, the maximum height is achieved when its final velocity is zero. Therefore, the kinetic energy at the maximum height is zero. This means that all the initial kinetic energy (obtained from the bow) is converted into potential energy:
Initial Kinetic Energy = Final Potential Energy
So, we can write the equation as:
(1/2) * m * (velocity)^2 = m * g * h
Rearranging the equation, we get:
velocity^2 = 2 * g * h
Now, let's calculate the maximum height reached by the arrow. We can use the kinematic equation:
Final Velocity (v) = Initial Velocity (u) + (acceleration * time)
Since the arrow is fired vertically upward, the final velocity is zero at the highest point. The initial velocity is given as 8.0 m/s. The acceleration is equal to the acceleration due to gravity (g). We can assume that it takes the arrow the same amount of time to reach the maximum height as it takes to fall back down. Therefore, the total time of flight is twice the time taken to reach the maximum height.
Setting the final velocity to zero, the equation becomes:
0 = 8.0 m/s - (9.8 m/s^2) * t
Simplifying, we get:
t = 8.0 m/s / (9.8 m/s^2)
Now, we can calculate the total time of flight:
Total Time of Flight (T) = 2 * t
Next, we can plug this value of time into the equation for maximum height (h) to calculate it.
Now that we have the maximum height reached by the arrow, we can substitute it back into the equation for potential energy to find the work done by the bow:
Work (W) = m * g * h
Finally, we can calculate the average force applied to the arrow by the bow using the equation:
Average Force (F) = Work (W) / Distance (d)
Where:
d = distance over which the force is applied (given as 19 cm)
By following these steps, you should be able to calculate the average force applied to the arrow by the bow.