In a physics lab experiment, a spring clamped to the table shoots a 18 g ball horizontally. When the spring is compressed 20 cm, the ball travels horizontally 4.9 m and lands on the floor 1.3 m below the point at which it left the spring. What is the spring constant?
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To find the spring constant, we need to use the concept of potential energy and Hooke's Law. Here's how you can calculate it:
1. Start by considering the potential energy stored in the compressed spring. The potential energy equation for a spring is given by:
Potential Energy = (1/2) * k * x^2
where k is the spring constant and x is the compression distance.
2. In this case, the ball is traveling horizontally, so the potential energy is converted into kinetic energy, which is responsible for the horizontal distance traveled. Therefore, the potential energy of the spring can be equated to the kinetic energy of the ball.
3. The kinetic energy equation for an object is given by:
Kinetic Energy = (1/2) * m * v^2
where m is the mass of the object and v is its velocity.
4. Since the ball lands on the floor 1.3 m below its starting point, we can use the equation of motion to find the initial velocity in the vertical direction. The equation is:
Δy = v0y * t + (1/2) * g * t^2
where Δy is the vertical displacement (1.3 m), v0y is the initial vertical velocity, t is the time of flight, and g is the acceleration due to gravity (-9.8 m/s^2).
5. The time of flight can be found using the equation:
t = √(2Δy / g)
6. Now, we know the horizontal distance traveled (4.9 m) and the time of flight (from step 5), so we can determine the horizontal velocity using the equation:
v = Δx / t
7. Finally, substitute the values obtained in step 6 into the kinetic energy equation (step 3) and solve for the spring constant (k).
By following these steps and plugging in the given values, you should be able to find the spring constant.