A projectile is launched from a compressed spring on an inclined plane. The spring constant of the spring is k= 10 N/m, the mass is 1kg and the plane is inclined 30 degrees to the horizontal. If the spring is compressed 15 cm what will be the velocity of the projectile when it is launched?
1/2 mv^2=1.2 k x^2
solve for v
To determine the velocity of the projectile when it is launched, we can use the principles of energy conservation.
1. First, let's find the potential energy stored in the compressed spring.
The potential energy stored in a spring is given by the formula:
Potential Energy (PE) = (1/2) * k * x^2,
where k is the spring constant and x is the displacement of the spring from its equilibrium position (compressed distance).
Given: spring constant k = 10 N/m and compressed distance x = 15 cm = 0.15 m.
Plugging in the values, we get:
PE = (1/2) * 10 * (0.15)^2 = 0.1125 J
2. Next, let's calculate the gravitational potential energy of the projectile.
Gravitational potential energy (PE) = mass * gravitational acceleration * height,
where the height is the vertical distance traveled by the projectile.
Given: mass = 1 kg and inclined plane at 30 degrees to the horizontal.
Since the inclined plane makes an angle of 30 degrees with the horizontal, the vertical height can be calculated as:
height = x * sin(angle)
height = 0.15 m * sin(30 degrees) ≈ 0.075 m.
Gravitational potential energy = 1 * 9.8 * 0.075 = 0.735 J
3. Now, let's find the total energy of the projectile just before it is launched.
The total energy is the sum of the potential energy stored in the compressed spring and the gravitational potential energy.
Total energy (E) = Potential Energy of the spring (PE) + Gravitational Potential Energy (PE)
E = 0.1125 J + 0.735 J = 0.8475 J
4. By the principle of energy conservation, the total energy just before launching is equal to the kinetic energy of the projectile when it is launched.
Kinetic Energy (KE) = Total energy (E)
KE = 0.8475 J
5. Kinetic energy can be calculated using the formula:
KE = (1/2) * mass * velocity^2
Since the mass of the projectile is given as 1 kg and the kinetic energy is known, we can rearrange the formula to solve for velocity:
velocity = √(2 * KE / mass)
Plugging in the values, we get:
velocity = √(2 * 0.8475 / 1) ≈ √1.695 ≈ 1.3 m/s
Therefore, the velocity of the projectile when launched is approximately 1.3 m/s.