A spring gun is made by compressing a spring in a tube and then latching the spring at the compressed position. A 4.97-g pellet is placed against the compressed and latched spring. The spring latches at a compression of [01] cm, and it takes a force of 9.12 N to compress the spring to that point. (a) If the gun is �red vertically, how fast (m/s) is the pellet moving when it loses contact with the spring? (b) To what maximum height (m) will the pellet rise? (as measured from the original latched position)
To solve this problem, we can use the principles of conservation of mechanical energy. The initial potential energy stored in the compressed spring is equal to the final kinetic energy of the pellet when it loses contact with the spring. We can use the following steps to find the answers to the given questions:
(a) To find the speed of the pellet when it loses contact with the spring, we need to calculate the potential energy stored in the compressed spring and then equate it to the kinetic energy of the pellet.
1. Convert the spring compression from centimeters to meters:
[01] cm = [01] cm * (1 m / 100 cm) = [01] m
2. Calculate the potential energy stored in the compressed spring:
Potential Energy (PE) = (1/2) * k * x^2
The force required to compress the spring is 9.12 N at a compression of [01] m, so we can calculate the spring constant (k) using Hooke's Law:
Force (F) = k * x
k = F / x
k = 9.12 N / [01] m = 9.12 N/m
Now, substitute the values into the potential energy formula:
PE = (1/2) * 9.12 N/m * ([01] m)^2
3. Set the potential energy equal to the kinetic energy of the pellet:
PE = KE
(1/2) * 9.12 N/m * ([01] m)^2 = (1/2) * m * v^2
Since the mass of the pellet is 4.97 g, convert it to kilograms:
m = 4.97 g * (1 kg / 1000 g)
Solve for the velocity of the pellet (v) by rearranging the equation:
v^2 = (9.12 N/m * ([01] m)^2) / (4.97 g * (1 kg / 1000 g))
v = √((9.12 N/m * ([01] m)^2) / (4.97 g * (1 kg / 1000 g)))
Calculate the value of v using the given values and unit conversions.
(b) To determine the maximum height the pellet will rise, we can use the principle of conservation of mechanical energy.
1. Calculate the potential energy at the maximum height:
Potential Energy (PE) = (1/2) * m * v^2, where v is the velocity obtained in part (a).
2. At maximum height, the kinetic energy is zero, so the potential energy at that point is equal to the gravitational potential energy:
PE = m * g * h_max, where g is the acceleration due to gravity (approximately 9.8 m/s^2) and h_max is the maximum height.
Now, equate the potential energy calculated in step 1 to the gravitational potential energy:
(1/2) * m * v^2 = m * g * h_max
Solve for h_max:
h_max = (1/2) * v^2 / g
Calculate the value of h_max using the obtained velocity and the given value of g.
By following these steps and substituting the given values into the equations, you can find the answers to parts (a) and (b) of the problem.