# To test out your new foam dart blaster, you hang a small cube, size 5cm on a side with mass 0.4...

## Question:

To test out your new foam dart blaster, you hang a small cube, size 5cm on a side with mass 0.4 kg, from your ceiling. You then fire a 46-gram foam dart at the bottom of it with a speed of 15.3 m/s, and the dart sticks to the cube. Calculate the height in m that the block will reach relative to where it started.

## Energy Conservation:

The energy conservation principle consists of a physical law that states that for closed systems, the total energy of the system should conserve. Energy is not created nor destroyed, it only transforms. This means that a system may undergo energy transformations, but the overall total energy of the system will remain constant. The total energy of a system can be determined by the sum of the total kinetic and potential energy of the system.

$$E_t=K_e+U_e\\ E_t=0.5mv^2+mgh $$

## Answer and Explanation:

In order to determine the maximum height of the pendulum block, we must use momentum conservation and energy conservation principles. We use momentum conservation to determine the speed of the pendulum after the block is impacted by the dart. After that, we use the energy conservation principle to determine the maximum height it is able to reach.

**Information:**

- Mass of the cube: {eq}m_c=0.4\ kg {/eq}

- Mass of the dart: {eq}m_d=0.046\ kg {/eq}

- Initial speed of the dart: {eq}v_d=15.3\ m/s {/eq}

- Acceleration due to gravity: {eq}g=9.8\ m/s^2 {/eq}

**Momentum conservation equation:**

{eq}m_dv_d=(m_c+m_d)v\\ v=\dfrac{m_dv_d}{m_c+m_d}\\ v=\dfrac{(0.046\ kg)(15.3\ m/s)}{0.4+0.046}\\ v=1.58\ m/s {/eq}

Finally, we use the energy conservation principle to determine the height the block is able to reach. Kinetic energy will completely convert to gravitational potential energy.

{eq}E_i=E_f\\ K_e=U_g\\ \dfrac{1}{2}mv^2=mgh\\ h=\dfrac{v^2}{2g}\\ h=\dfrac{(1.58\ m/s)^2}{2(9.8\ m/s^2)}\\ h=0.127\ m {/eq}

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