pre - calculus worksheet forces on objects and work name: ______ period: ______ the first 2 problem…

pre - calculus worksheet forces on objects and work name: ______ period: ______ the first 2 problem situations are like example 1 in your notes. you must show work 1. a golfer hits a golf ball due east at 50 mph. however, a wind is blowing from the south at 12 mph. a) write the components of the resultant vector. 1 b) what is the actual speed of the ball with the 12 - mph wind? (speed is magnitude.) 2 c) what is the angle the balls path makes with the horizontal? 3 2. a hot air balloon rises at a rate of 2.73 mph, and the wind is blowing at 5.12 mph from the west. a) write the components of the resultant vector 4 b) how fast is the balloon really going? (speed is magnitude.) 5 c) what angle does its path make with the ground? 6

pre - calculus worksheet forces on objects and work name: ______ period: ______ the first 2 problem situations are like example 1 in your notes. you must show work 1. a golfer hits a golf ball due east at 50 mph. however, a wind is blowing from the south at 12 mph. a) write the components of the resultant vector. 1 b) what is the actual speed of the ball with the 12 - mph wind? (speed is magnitude.) 2 c) what is the angle the balls path makes with the horizontal? 3 2. a hot air balloon rises at a rate of 2.73 mph, and the wind is blowing at 5.12 mph from the west. a) write the components of the resultant vector 4 b) how fast is the balloon really going? (speed is magnitude.) 5 c) what angle does its path make with the ground? 6

Answer

Explanation:

Step1: Analyze vector components for golf - ball problem

Let the east - west direction be along the x - axis (east is positive x) and north - south direction be along the y - axis (north is positive y). The velocity of the golf ball due east is $\vec{v}{ball}=(50,0)$ and the velocity of the wind from the south is $\vec{v}{wind}=(0,12)$. The resultant vector $\vec{R}=\vec{v}{ball}+\vec{v}{wind}=(50,12)$.

Step2: Calculate the magnitude of the resultant vector for golf - ball

The magnitude of a vector $\vec{R}=(x,y)$ is given by $|\vec{R}|=\sqrt{x^{2}+y^{2}}$. For $\vec{R}=(50,12)$, $|\vec{R}|=\sqrt{50^{2}+12^{2}}=\sqrt{2500 + 144}=\sqrt{2644}\approx51.42$ mph.

Step3: Calculate the angle of the resultant vector for golf - ball

The angle $\theta$ that the vector $\vec{R}=(x,y)$ makes with the horizontal is given by $\tan\theta=\frac{y}{x}$. For $\vec{R}=(50,12)$, $\tan\theta=\frac{12}{50}=0.24$, so $\theta=\arctan(0.24)\approx13.5^{\circ}$.

Step4: Analyze vector components for hot - air balloon problem

Let the east - west direction be along the x - axis (east is positive x) and up - down direction be along the y - axis (up is positive y). The velocity of the hot - air balloon rising is $\vec{v}{balloon}=(0,2.73)$ and the velocity of the wind from the west is $\vec{v}{wind}=(- 5.12,0)$. The resultant vector $\vec{R}=\vec{v}{balloon}+\vec{v}{wind}=(-5.12,2.73)$.

Step5: Calculate the magnitude of the resultant vector for hot - air balloon

Using the magnitude formula $|\vec{R}|=\sqrt{x^{2}+y^{2}}$, for $\vec{R}=(-5.12,2.73)$, $|\vec{R}|=\sqrt{(-5.12)^{2}+2.73^{2}}=\sqrt{26.2144 + 7.4529}=\sqrt{33.6673}\approx5.80$ mph.

Step6: Calculate the angle of the resultant vector for hot - air balloon

Using $\tan\theta=\frac{y}{x}$, for $\vec{R}=(-5.12,2.73)$, $\tan\theta=\frac{2.73}{-5.12}\approx - 0.533$. Since $x<0$ and $y > 0$, $\theta=\pi+\arctan(-0.533)\approx180^{\circ}-28.1^{\circ}=151.9^{\circ}$ (measured counter - clockwise from the positive x - axis) or $\theta\approx90^{\circ}+\arctan(\frac{5.12}{2.73})\approx151.9^{\circ}$

Answer:

  1. A. $(50,12)$ B. Approximately $51.42$ mph C. Approximately $13.5^{\circ}$
  2. A. $(-5.12,2.73)$ B. Approximately $5.80$ mph C. Approximately $151.9^{\circ}$