Question

Mickey, a daredevil mouse of mass of .0217 kg, is attempting to become the world's first "mouse cannonball." He is loaded into a spring-powered gun pointing up at some angle and is shot into the air. The gun's spring has a force constant of 78.9 N/m and is initially compressed a distance of .135 m from its relaxed position.

If Mickey has a constant horizontal speed of 2.25 m/s while he is flying through the air, how high h above his initial location in the gun does Mickey soar? Assume g = 9.81 m/s^2.

Answer 1

Answer 2

 Calculate the initial elastic potential energy stored in the spring:

$$U_{\text{spring}}=\frac{1}{2}k{x}^2=\frac{1}{2}\times 78.9\times (0.135{)}^2=0.719\text{J}$$

1. Determine the total initial kinetic energy when Mickey is launched:
   The spring's potential energy converts entirely into kinetic energy (assuming no energy losses):

   $$K_{\text{total}}=U_{\text{spring}}=0.719\text{J}$$

2. Calculate the horizontal component of the kinetic energy:

   $$K_x=\frac{1}{2}m{v}_x^2=\frac{1}{2}\times 0.0217\times (2.25{)}^2=0.055\text{J}$$

3. $$=0.055\text{J}$$

4. Find the vertical component of the kinetic energy:
   The total kinetic energy is the sum of horizontal and vertical components:

   $$K_y=K_{\text{total}}-K_x=0.719-0.055=0.664\text{J}$$

5. $$=0.664\text{J}$$

6. Compute the initial vertical velocity ($v_y$) using $K_y$:

   $$K_y=\frac{1}{2}m{v}_y^2⟹v_y=\sqrt{\frac{2K_y}{m}}=\sqrt{\frac{2\times 0.664}{0.0217}}=7.83\text{m/s}$$

7. $$=7.83\text{m/s}$$

8. Determine the maximum height ($h$) using the vertical motion equation:
   At the highest point, the vertical velocity becomes zero. Using:

   $$v_y^2=2gh⟹h=\frac{v_y^2}{2g}=\frac{(7.83{)}^2}{2\times 9.81}=3.12\text{m}$$

9. $$=3.12\text{m}$$

Mickey soars 3.12 meters above his initial location in the gun.

Answer 3

Let's break down this problem step-by-step:

1. Understand the Energy Conversion

The initial potential energy stored in the compressed spring is converted into kinetic energy of Mickey when he is launched. This kinetic energy is then converted into gravitational potential energy as Mickey rises in the air.

2. Calculate the Initial Potential Energy of the Spring

The potential energy (PE) stored in a spring is given by:

PE = (1/2) * k * x²

where:

• k is the spring constant (78.9 N/m)

• x is the compression distance (0.135 m)

PE = (1/2) * 78.9 N/m * (0.135 m)²

PE ≈ 0.7188 J

3. Calculate Mickey's Initial Kinetic Energy

Assuming all the spring's potential energy is converted into Mickey's kinetic energy (KE) at launch, then:

KE = PE ≈ 0.7188 J

4. Determine Mickey's Launch Velocity

The kinetic energy is also given by:

KE = (1/2) * m * v²

where:

• m is Mickey's mass (0.0217 kg)

• v is Mickey's launch velocity

0.7188 J = (1/2) * 0.0217 kg * v²

v² = (2 * 0.7188 J) / 0.0217 kg

v² ≈ 66.1152 m²/s²

v ≈ √66.1152 m²/s²

v ≈ 8.131 m/s

5. Find the Vertical Component of Mickey's Launch Velocity

We know Mickey has a constant horizontal speed of 2.25 m/s. Let's call the launch angle θ.  We have:

v_horizontal = v * cos(θ)

v_vertical = v * sin(θ)

We also know:

v² = v_horizontal² + v_vertical²

(8.131 m/s)² = (2.25 m/s)² + v_vertical²

66.1152 = 5.0625 + v_vertical²

v_vertical² = 61.0527

v_vertical ≈ 7.8136 m/s

6. Calculate the Maximum Height (h)

We can use the following kinematic equation to find the maximum height:

v_final² = v_initial² + 2 * a * Δy

where:

• v_final is the final vertical velocity (0 m/s at the highest point)

• v_initial is the initial vertical velocity (7.8136 m/s)

• a is the acceleration due to gravity (-9.81 m/s², negative because it's downward)

• Δy is the vertical displacement (the height h we want to find)

0 = (7.8136 m/s)² + 2 * (-9.81 m/s²) * h

0 = 61.0527 - 19.62 * h

h = 61.0527 / 19.62

h ≈ 3.111 m

Answer:

Mickey soars approximately 3.11 meters above his initial location in the gun.