In physics lab experiment, a compressed spring launches a 20g metal ball at a 30 degrees angle. Compressed the 20cm causes the ball to hit the floor 1.5m below the point at which it leaves the spring after travelling 5.0m horizontal. What is the spring constant?

Question

In physics lab experiment, a compressed spring launches a 20g metal ball at a 30 degrees angle. Compressed the 20cm causes the ball to hit the floor 1.5m below the point at which it leaves the spring after travelling 5.0m horizontal. What is the spring constant?

Answer 1

To find the spring constant, we can use the principles of projectile motion and the conservation of mechanical energy.

First, let's calculate the initial velocity of the ball. We know that the ball is launched at a 30-degree angle, and its horizontal range is 5.0m. The horizontal component of the initial velocity can be found using the formula:

Vx = V * cos(theta)

Here, V represents the magnitude of the initial velocity, and theta is the launch angle.

Vx = V * cos(30°)
Vx = V * sqrt(3)/2

Since the horizontal range is the same as the horizontal component of the velocity multiplied by the time of flight, we can set up the equation:

5.0m = (V * sqrt(3)/2) * t

Next, let's calculate the vertical component of the initial velocity using the height the ball falls below the launch point. The vertical distance can be calculated using the formula for free fall motion:

height = (1/2) * g * t^2

Here, g is the acceleration due to gravity (approximately 9.8 m/s^2), and t is the time of flight.

Plugging in the values, we have:

1.5m = (1/2) * 9.8 m/s^2 * t^2

Now, let's consider the conservation of mechanical energy. At the highest point in the ball's trajectory, its potential energy is maximized, and at the launch point, its potential energy is zero. The initial potential energy is converted into kinetic energy, given by:

PE_initial = KE_final

Since the potential energy is proportional to the displacement of the spring (x) and the spring constant (k), we can rewrite the equation as:

(1/2) * k * x^2 = (1/2) * m * V^2

Here, k is the spring constant, x is the compression distance, and m is the mass of the ball.

Plugging in the known values:

(1/2) * k * (0.20m)^2 = (1/2) * 0.020kg * V^2

Simplifying the equation, we find:

k = (m * V^2) / x^2

Finally, substitute the values:

k = (0.020kg * ((V * sqrt(3)/2) * t)^2) / (0.20m)^2

Now, plug in the values for V and t obtained earlier:

k = (0.020kg * (((5.0m / t) * sqrt(3))/2)^2) / (0.20m)^2

After substituting the values, you can calculate the spring constant (k) by solving the equation.