# Sue and Jenny kick a soccer ball at exactly the same time. Sue's foot exerts a force of 40.6 N to...

## Question:

Sue and Jenny kick a soccer ball at exactly the same time. Sue's foot exerts a force of 40.6 N to the north. Jenny's foot exerts a force of 84.8 N to the east.

a) What is the magnitude of the resultant force on the ball? Answer in units of N.

b) What is the direction of the resultant force (measured from the East)? Answer in units of N.

## Vector Magnitude & Direction:

When two people apply forces on a ball and the forces are northward and eastward i.e. perpendicular, the resultant is the vector sum of the two forces and since the forces are perpendicular, the vector sum is the Pythagorean Hypotenuse of the two.

## Answer and Explanation:

#### Part a)

Sue and jenny kick the ball in perpendicular directions. Sue applies a force {eq}F_y = 40.6\ \text{N} {/eq} northwards and Jenny applies a force {eq}F_x = 84.8\ \text{N} {/eq} eastward. because the two are perpendicular, the resultant F is the Pythagorean hypotenuse of the two, i.e.

{eq}\begin{align*} F &= \sqrt{F_y^2+F_x^2}\\ &= \sqrt{(40.6\ \text{N})^2+(84.8\ \text{N})^2}\\ &\color{blue}{\approx \boxed{94.0\ \text{N}}} \end{align*} {/eq}

#### Part b)

To determine the direction {eq}\theta {/eq} north of east, we use a suitable trigonometric ratio, i.e.

{eq}\begin{align*} \tan\theta &= \frac{F_y}{F_x}\\ \theta &= \arctan{\frac{40.6\ \text{N}}{84.8\ \text{N}}}\\ &\color{blue}{\approx \boxed{25.6\ ^\circ}} \end{align*} {/eq}