Consider an archery bow as behaving like a spring with a spring constant of 493 N/m. If an archer pulls the bowstring back for a distance of 42 cm, what is the elastic potential energy of the "loaded" bow?


If the mass of the arrow is 39 grams, what is its speed as it leaves the bow?

During a free kick at a soccer playoff game a 0.5 kg ball at the peak of the kick 14.7 m above the field had a kinetic energy of 34.0 J. How fast was the ball booted by the player on the field?

To calculate the elastic potential energy of the loaded bow, we need to know the spring constant and the displacement. The elastic potential energy (PE) in a spring is given by the formula:

PE = (1/2) * k * x^2

Where PE is the elastic potential energy, k is the spring constant, and x is the displacement.

Given that the spring constant is 493 N/m and the displacement is 42 cm, we first need to convert the displacement from cm to meters by dividing by 100:

x = 42 cm / 100 = 0.42 m

Now we can calculate the elastic potential energy:

PE = (1/2) * k * x^2
= (1/2) * 493 N/m * (0.42 m)^2
= 0.5 * 493 N/m * 0.1764 m^2
= 43.53 J (rounded to two decimal places)

Therefore, the elastic potential energy of the loaded bow is 43.53 Joules.

Now, to determine the speed of the arrow as it leaves the bow, we can use the principle of conservation of energy. The potential energy stored in the bow is converted into kinetic energy of the arrow.

The elastic potential energy we just calculated (43.53 J) is equal to the kinetic energy (KE) of the arrow:

KE = 43.53 J

The kinetic energy of an object is given by the formula:

KE = (1/2) * m * v^2

Where KE is the kinetic energy, m is the mass of the arrow, and v is the velocity (speed) of the arrow.

Given that the mass of the arrow is 39 grams, we need to convert it to kilograms by dividing by 1000:

m = 39 g / 1000 = 0.039 kg

Now we can solve for the velocity:

43.53 J = (1/2) * 0.039 kg * v^2
v^2 = (43.53 J * 2) / 0.039 kg
v^2 = 1118.46 m^2/s^2
v = √(1118.46) m/s
v ≈ 33.45 m/s (rounded to two decimal places)

Therefore, the speed of the arrow as it leaves the bow is approximately 33.45 m/s.