Chapter 9, Problem 57A

To determine

The magnitude and direction of the momentum of the bale the moment before the plane strikes the ground.

Answer to Problem 57A

The magnitude of the momentum of the bale the moment before the plane strikes the ground is $p=887.5\text{kg}\cdot \text{m}/\text{s}$ .

The direction of the momentum of the bale the moment before the plane strikes the ground is $\theta =43.6°$ .

Explanation of Solution

Given:

The initial velocity of the plane is $v_i=36.0\text{m}/\text{s}$ .

The altitude is $s=60.0\text{ m}$ .

The weight of the bale of hay is $175\text{ N}$ .

Formula used:

The expression for the velocity of the bale before striking the ground $\left(v\right)$ is,

  $v^2=v_i^2+2gs$

Here, $v_i$ is the initial velocity, $g$ is the acceleration due to gravity, and $s$ is the altitude.

The expression for the magnitude of the momentum is,

  $p=\frac{F_{g}v}{g}$

Here, $F_g$ is the weight of the bale of hay and $v$ is the velocity of the bale before striking the ground.

The expression for the direction of the momentum is,

  $\tan \theta =\frac{\sqrt{2gs}}{v_i}$

Calculation:

The velocity of the bale before striking the ground as follows:

  $\begin{array}{l}v=\sqrt{v_i^2+2gs}\\ v=\sqrt{{\left(36.0\text{m}/\text{s}\right)}^2+2\times \left(9.8\text{m}/{\text{s}}^2\right)\times \left(60.0\text{ m}\right)}\\ v=\sqrt{2,472}\\ v=49.7\text{m}/\text{s}\end{array}$

The magnitude of the momentum is,

  $\begin{array}{l}p=\frac{F_{g}v}{g}\\ p=\frac{\left(175\text{ N}\times \frac{1\text{ kg}\cdot \text{m}}{1\text{ N}}\right)\times \left(49.7\text{m}/\text{s}\right)}{\left(\text{9}\text{.8 }\text{m}/{\text{s}}^2\right)}\\ p=887.5\text{kg}\cdot \text{m}/\text{s}\end{array}$

The direction of the momentum as follows:

  $\begin{array}{l}\tan \theta =\frac{\sqrt{2gs}}{v_i}\\ \tan \theta =\frac{\sqrt{2\times \left(9.8\text{m}/{\text{s}}^2\right)\times \left(60.0\text{ m}\right)}}{\left(36.0\text{m}/\text{s}\right)}\\ \theta ={\tan }^{-1}\left(0.9525\right)\\ \theta =43.6°\end{array}$

Conclusion:

Therefore, the magnitude of the momentum of the bale the moment before the plane strikes the ground is $p=887.5\text{kg}\cdot \text{m}/\text{s}$ and the direction of the momentum of the bale the moment before the plane strikes the ground is $\theta =43.6°$ .