A spring of force constant 655 N/m is used to launch a 5.25 kg mass from the edge of a frictionless table that is 1.20 m above the floor. How far horizontally away from the table will the mass land if the spring was compressed 42.0 cm before launching?
To determine how far horizontally the mass will land, we need to analyze the vertical motion of the mass.
The potential energy stored in the compressed spring gets converted into the gravitational potential energy of the mass as it's launched.
First, let's find the potential energy stored in the compressed spring using Hooke's law:
Potential energy of a spring = (1/2) * k * x^2
where k is the force constant of the spring and x is the compression of the spring.
Plugging in the values given:
Potential energy of the spring = (1/2) * 655 N/m * (0.42 m)^2
= 56.7 J
This potential energy is then converted into gravitational potential energy as the mass is lifted. The gravitational potential energy is given by:
Gravitational potential energy = mass * g * height
where mass is the mass of the object, g is the acceleration due to gravity (9.8 m/s^2), and height is the vertical distance above the floor.
Plugging in the values:
Gravitational potential energy = 5.25 kg * 9.8 m/s^2 * 1.20 m
= 61.74 J
Since the potential energy of the spring is equal to the gravitational potential energy, we can equate the two equations:
Potential energy of the spring = Gravitational potential energy
This gives us:
(1/2) * 655 N/m * (0.42 m)^2 = 5.25 kg * 9.8 m/s^2 * 1.20 m
Now, let's solve for the total distance traveled horizontally (d). Since there is no horizontal force acting on the mass, the horizontal motion is independent of the vertical motion.
We can use the equation of motion for constant acceleration to find the time of flight (t) from the table to the floor:
Height = (1/2) * g * t^2
Plugging in the values:
1.20 m = (1/2) * 9.8 m/s^2 * t^2
Solving for t:
t^2 = (2 * 1.20 m) / 9.8 m/s^2
t^2 = 0.2449 s^2
t = √(0.2449 s^2)
t ≈ 0.4949 s
Now, we can use the time of flight to find the horizontal distance traveled:
Distance = velocity * time
The initial velocity in the horizontal direction (v₀x) is 0 because the mass is launched horizontally.
Thus, the horizontal distance traveled (d) can be found as:
d = v₀x * t
= 0 * 0.4949 s
= 0
Therefore, the mass will land at 0 meters horizontally away from the table.