Robin Hood pulls the string of his bow backwards 0.6 m with a force of 170 N. The mass of the arrow is 0.095 kg. If the bow is 92% efficient, at what speed will the arrow leave
the bow?
PE = mgd 0.92= 0=.095*9.8*0.6*0.92 = __N. = KE
KE = PE = 0.5 MV^2 = 0.5*0.095V^2
V =
To calculate the speed at which the arrow will leave the bow, we need to determine the amount of work done on the arrow and use it to calculate the kinetic energy.
First, let's calculate the work done on the arrow. The work done is equal to the force applied multiplied by the distance over which the force is applied. In this case, the force is the tension in the bowstring and the distance is the displacement of the bowstring.
Work = Force * Distance
Given:
Force (F) = 170 N
Distance (d) = 0.6 m
Work = 170 N * 0.6 m
Next, we need to determine the efficiency of the bow. Efficiency is defined as the ratio of useful work output to the total work input.
Efficiency = Useful work output / Total work input
Given:
Efficiency = 92% = 0.92
To find the total work input, we can use the equation:
Total work input = Useful work output / Efficiency
Plugging in the values, we get:
Total work input = Work / Efficiency
Total work input = (170 N * 0.6 m) / 0.92
Now we can proceed to calculate the work done on the arrow (useful work output) by multiplying the total work input by the efficiency:
Useful work output = Total work input * Efficiency
Next, we can calculate the kinetic energy of the arrow using the work-energy principle, which states that the work done on an object is equal to its change in kinetic energy.
Kinetic Energy = Useful work output
Now we have the kinetic energy, and we can use it to find the velocity of the arrow.
The kinetic energy of an object can be calculated using the equation:
Kinetic Energy = 0.5 * mass * velocity^2
Given:
Mass (m) = 0.095 kg
Kinetic Energy = Useful work output (calculated in the previous step)
Plugging in the values, we get:
Kinetic Energy = 0.5 * 0.095 kg * velocity^2
Setting the two equations for kinetic energy equal to each other, we can solve for velocity:
0.5 * 0.095 kg * velocity^2 = Useful work output
Simplifying:
velocity^2 = (Useful work output) / (0.5 * 0.095 kg)
Now we can solve for velocity by taking the square root of both sides:
velocity = √[(Useful work output) / (0.5 * 0.095 kg)]
By plugging in the values calculated earlier for the useful work output, we can determine the final velocity of the arrow.