a popgun uses a spring for which k = 20 N/cm. when cocked, the spring is compressed 3.0 cm. how high can the gun shoot a 5.0-g projectile

Jiskha software automatically censors some words, but I think I know what you mean.

(It rhymes with clocked)

The stored potential in the spring when loaded will be (1/2)kX^2 = (1/2)*20*9 = 90 N-cm = 0.9 Joules

Set 0.90 J = (1/2)M V^2 and solve for V

V = sqrt(2*0.90/.005) = 19.0 m/s

Thanks a lot!

To determine the height at which the popgun can shoot the projectile, we need to calculate the potential energy stored in the compressed spring.

Given:

Spring constant, k = 20 N/cm = 20 N/0.01 m = 2000 N/m
Compression distance of the spring, x = 3.0 cm = 0.03 m
Mass of the projectile, m = 5.0 g = 0.005 kg

Step 1: Calculate the potential energy stored in the spring.

The formula for potential energy stored in a spring is given by:

Potential energy (PE) = (1/2) * k * x^2

Plugging in the values:

PE = (1/2) * 2000 N/m * (0.03 m)^2
= (1/2) * 2000 N/m * 0.0009 m^2
= 0.9 J

Step 2: Calculate the maximum height the projectile can reach.

At the maximum height, all of the potential energy stored in the spring is converted to gravitational potential energy of the projectile.

Gravitational potential energy (GPE) = m * g * h

Here, g is the acceleration due to gravity (approximately 9.8 m/s^2) and h is the maximum height.

Plugging in the values:

0.9 J = 0.005 kg * 9.8 m/s^2 * h
h = 0.9 J / (0.005 kg * 9.8 m/s^2)
h ≈ 18.37 m

Therefore, the popgun can shoot the 5.0 g projectile to a maximum height of approximately 18.37 meters.

To determine how high the gun can shoot a 5.0-g projectile, we need to use the potential energy stored in the compressed spring and convert it to kinetic energy, assuming no energy losses.

First, let's recall the formula for the potential energy stored in a spring:

Potential Energy (U) = (1/2) * k * x^2

Where:
- k is the spring constant (also known as the stiffness) in N/cm.
- x is the displacement or compression of the spring in cm.

In this case, k = 20 N/cm and x = 3.0 cm.

Now, plug these values into the formula to find the potential energy:

U = (1/2) * (20 N/cm) * (3.0 cm)^2
U = (1/2) * (20 N/cm) * (9.0 cm^2)
U = 90 N.cm

Next, we need to convert this potential energy to kinetic energy. The equation for kinetic energy is as follows:

Kinetic Energy (K) = (1/2) * m * v^2

Where:
- m is the mass of the projectile in kg.
- v is the velocity of the projectile in m/s.

In this case, we need to find the velocity at which the projectile is shot. To do that, we'll equate the potential energy to the kinetic energy:

U = K

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

Since we know m = 5.0 g = 0.005 kg, we can solve for v:

(1/2) * (20 N/cm) * (3.0 cm)^2 = (1/2) * (0.005 kg) * v^2
90 N.cm = 0.0025 kg * v^2
v^2 = (90 N.cm) / (0.0025 kg)
v^2 = 36000 (m^2/s^2)

v = sqrt(36000) m/s
v ≈ 189.74 m/s

Once we know the velocity, we can calculate the maximum height reached by the projectile using the formula for vertical projectile motion. Since we know that the vertical velocity at its maximum height is 0 m/s (assuming no air resistance), we can use the following equation:

v^2 = u^2 + 2a * s

Where:
- v is the final velocity (0 m/s at maximum height)
- u is the initial velocity (189.74 m/s in the upward direction)
- a is the acceleration due to gravity (-9.8 m/s^2)
- s is the maximum height we need to find.

Rearranging the equation, we have:

0^2 = (189.74 m/s)^2 + 2 * (-9.8 m/s^2) * s
0 = 189.74^2 - 19.6s
19.6s = 189.74^2
s = (189.74^2) / 19.6

Using a calculator, evaluate (189.74^2) / 19.6, which gives approximately 1880.45m.

Therefore, the maximum height the gun can shoot a 5.0-g projectile is approximately 1880.45 meters.