Question

1. a guitar string is about 50 cm long, measured to the point of strumming. playing middle c at forte volume causes it to move back and forth, making 3° angles with its original position. how far did the guitarist move the string sideways to achieve this note and volume? 11. a pirate in the middle of a sea battle sets his cannon to a 20° angle and launches a cannonball at 200 m/s toward the merchant ship. how quickly is the cannonball going to be covering the space between ships?

Explanation:

Step1: Identify right - triangle relationship for guitar string

We have a right - triangle where the length of the guitar string is the hypotenuse ($l = 50$ cm) and the angle between the original and deflected position of the string is $\theta=3^{\circ}$. We want to find the side opposite the angle $\theta$, let's call it $x$. We use the sine function $\sin\theta=\frac{x}{l}$.
$x = l\times\sin\theta$

Step2: Substitute values

Substitute $l = 50$ cm and $\theta = 3^{\circ}$ into the formula. Since $\sin(3^{\circ})\approx0.0523$, then $x=50\times0.0523 = 2.615$ cm.

Step3: Identify horizontal - velocity formula for cannonball

For the cannonball, if the initial velocity is $v_0 = 200$ m/s and the launch angle is $\theta = 20^{\circ}$, the horizontal component of the velocity $v_x$ gives the rate at which the cannonball covers the distance between the ships. The formula for the horizontal component of velocity is $v_x=v_0\cos\theta$.

Step4: Substitute values

Substitute $v_0 = 200$ m/s and $\theta = 20^{\circ}$ into the formula. Since $\cos(20^{\circ})\approx0.9397$, then $v_x=200\times0.9397 = 187.94$ m/s.

Answer:

1. The guitarist moved the string sideways by approximately $2.62$ cm.
2. The cannonball is covering the space between ships at approximately $187.94$ m/s.