An archer puts a 0.34 kg arrow to the bowstring. An average force of 213.1 N is exerted
to draw the string back 1.48 m.
The acceleration of gravity is 9.8 m/s
2
.
Assuming no frictional loss, with what
speed does the arrow leave the bow?
Answer in units of m/s
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To find the speed at which the arrow leaves the bow, we can use the principle of conservation of energy. The potential energy stored in the bow is converted into the kinetic energy of the arrow as it is released.
First, let's calculate the potential energy stored in the bowstring. The potential energy is equal to the work done to draw the string back.
Potential energy (PE) = Work done
The work done can be calculated using the formula:
Work = Force × Distance
Given that the force is 213.1 N and the distance is 1.48 m, we can calculate the work done:
Work = 213.1 N × 1.48 m
Next, we'll calculate the potential energy:
Potential energy (PE) = Work done
Using the formula for potential energy:
PE = mgh
Where m is the mass of the arrow, g is the acceleration due to gravity, and h is the height through which the arrow was raised.
Since the arrow is raised vertically, the height is equal to the distance the string was drawn back, which is 1.48 m. Also, the mass of the arrow is given as 0.34 kg, and the acceleration due to gravity is 9.8 m/s².
Now, we can calculate the potential energy:
PE = 0.34 kg × 9.8 m/s² × 1.48 m
The potential energy obtained will be in joules (J).
Next, we'll calculate the kinetic energy of the arrow when it is released, which is equal to the potential energy stored in the bow.
Kinetic energy (KE) = Potential energy (PE)
Finally, we'll use the formula for kinetic energy to find the speed of the arrow:
KE = (1/2)mv²
Where m is the mass of the arrow and v is the speed of the arrow.
Rearranging the formula to solve for v:
v = √((2KE) / m)
Substituting the value of KE from the potential energy calculated earlier:
v = √((2 × PE) / m)
Now, we can calculate the speed of the arrow:
v = √((2 × potential energy) / mass)
Plug in the values and calculate to find the speed of the arrow in meters per second (m/s).