A spring-loaded toy gun is used to shoot a ball of mass m= 1.50 kg straight up in the air. The spring has spring constant k = 667 N/m. If the spring is compressed a distance of 25.0 cm from its equilibrium position y=0 and then released, the ball reaches a maximum height h_max (measured from the equilibrium position of the spring). There is no air resistance, and the ball never touches the inside of the gun. Assume that all movement occurs in a straight line up and down along the y axis. Find v_m the muzzle velocity of the ball (i.e., the velocity of the ball at the spring's equilibrium position y=0). I just don't understand where to begin with this question
To find the muzzle velocity (v_m) of the ball, we can use the principles of conservation of energy.
Step 1: Determine the potential energy stored in the spring when it is compressed:
The potential energy stored in a spring is given by the formula: PE = (1/2) k x^2, where k is the spring constant and x is the distance the spring is compressed.
In this case, the spring constant is k = 667 N/m and the distance compressed is 25.0 cm = 0.25 m. Therefore, the potential energy stored in the spring is:
PE = (1/2) * 667 * (0.25^2) = 20.92 Joules.
Step 2: Convert the potential energy to kinetic energy at the equilibrium position (y=0):
At the equilibrium position, all the potential energy of the spring is converted to kinetic energy of the ball. According to conservation of energy, PE = KE.
Therefore, the kinetic energy at the equilibrium position is equal to the potential energy stored in the spring: KE = 20.92 Joules.
Step 3: Find the muzzle velocity using the kinetic energy equation:
The kinetic energy of an object can be expressed as: KE = (1/2) m v^2, where m is the mass of the ball and v is its velocity.
In this case, the mass of the ball is given as m = 1.50 kg. Therefore, we have:
(1/2) * 1.50 * v_m^2 = 20.92
Rearranging the equation and solving for v_m, we have:
v_m = sqrt(2 * 20.92 / 1.50)
v_m = sqrt(27.87)
v_m ≈ 5.28 m/s
Therefore, the muzzle velocity of the ball is approximately 5.28 m/s.
To find the muzzle velocity of the ball, we need to use the principle of conservation of mechanical energy. The energy stored in the compressed spring is converted into the kinetic energy of the ball at the spring's equilibrium position.
Here's how you can proceed:
1. First, let's find the potential energy stored in the compressed spring. The potential energy of a spring is given by the equation: PE = (1/2)kx², where k is the spring constant and x is the displacement from the equilibrium position.
In this case, k = 667 N/m and x = 25.0 cm = 0.25 m, so the potential energy stored in the spring is: PE = (1/2)(667 N/m)(0.25 m)².
2. The potential energy stored in the spring is converted into the kinetic energy of the ball at the spring's equilibrium position. The kinetic energy of an object is given by the equation: KE = (1/2)mv², where m is the mass of the ball and v is its velocity.
In this case, the potential energy stored in the spring equals the kinetic energy of the ball at the spring's equilibrium position: (1/2)(667 N/m)(0.25 m)² = (1/2)(1.50 kg)v².
3. Now, we can calculate the muzzle velocity of the ball by rearranging the equation:
v² = [(2)(667 N/m)(0.25 m)²] / (1.50 kg).
Solve this equation for v.
4. Finally, take the square root of the calculated value of v² to get the muzzle velocity v.
Follow these steps to find the value of v.
M*g* h_max = (1/2) M (v_m)^2
M cancels out.
Solve for the initial velocity, v_m .