Question

In Fig. 9-78, a3.2 kgbox of running shoes slides on a horizontal frictionless table and collides with a 2.0 kg box of ballet slippers initially at rest on the edge of the table, at height h = 0.40 mThe speed of the 3.2 kg box is 3.0 m/sjust before the collision. If the two boxes stick together because of packing tape on their sides, what is their kinetic energy just before they strike the floor?

Short answer

The kinetic energy of the boxes just before they strike the floor, $K{E}_f$ is 29 J .

Step-by-step solution

Understanding the given information

i) Mass of running shoe box, $m_{1}is3.2kg$.

ii) Mass of ballet slipper box, $m_{2}is2.0kg$.

iii) Height of the table, $h_{i}is0.40m$.

iv) Speed of the mass $m_1$, $V_{i1}is3.0m/s$.

Concept and formula used in the given question

You can use the concept of conservation of momentum and conservation of mechanical energy. Using conservation of momentum, you find the initial velocity of the boxes at the edge of the table. Using conservation of mechanical energy, you can find the final velocity at the ground

$$\begin{gathered} m_1{V}_{i1}+m_2{V}_{i2}=m_1{V}_{f1}+m_2{V}_{f2} \\ \frac{1}{2}m{v}_i^2+mg{h}_1=\frac{1}{2}m{v}_f^2+mg{h}_2 \end{gathered}$$ $$\begin{gathered} m_1{V}_{i1}+m_2{V}_{12}=m_1{V}_{f1}+m_2{V}_{f2} \\ \frac{1}{2}m{v}_i^2+mg{h}_1=\frac{1}{2}m{v}_f^2+mg{h}_2 \end{gathered}$$

Calculation for the kinetic energy when two boxes stick together because of packing tape on their sides

First, we find the velocity of theboxes at the edge of the table; both move together, so they have thesame velocity, v; we can say that v can be calculated using momentum conservation law as,

$$\begin{gathered} \left(3.2\right)\left(3.0\right)+\left(20\right)\left(0\right)=\left(3.2\right)v+\left(2.0\right)v \\ v=\frac{9.6}{5.2} \\ =1.846m/s \end{gathered}$$

Now, this is the initial velocity for thenext projectile motion. We know that both boxes initially have kinetic energy and potential energy, which will convert to kinetic energy at the ground.

You get,

$$\begin{gathered} K{E}_i+P{E}_i=K{E}_f+P{E}_f \\ \frac{1}{2}m{v}^2+mg{h}_1=K{E}_f+mg{h}_f \end{gathered}$$

Substitute the values in the above expression, and we get,

$$\begin{gathered} \frac{1}{2}\left(5.2kg\right)\left(1.846m/s^2\right)+\left(5.2kg\right)\left(9.8m/s^2\right)\left(0.40m\right)=K{E}_f+\left(5.2\right)\left(9.8\right)\left(0\right) \\ K{E}_f=29.24\left(1kg.m^2/s^2\times \frac{1J}{1kg.m^2/s^2}\right) \\ =29.24J \\ =29J \end{gathered}$$

The kinetic energy of the boxes just before they strike the floor, $K{E}_{f}is29J$.