Chapter 2: Problem 7

At one moment in a football game, player A exerts a force to the east on player \(\mathrm{B}\). At the same time, a teammate of \(\mathrm{A}\) exerts the samesized force to the south on player \(\mathrm{B}\). In what direction is \(\mathrm{B}\) likely to go because of these forces?

Short Answer

Expert verified

Answer: Player B is likely to move in the direction that is 45 degrees south of east.

Step by step solution

Representing force vectors

Let the force exerted by player A to the east be represented by vector \(\vec{F_A}\) and the force exerted by A's teammate to the south be represented by vector \(\vec{F_B}\). Since both forces have the same magnitude, we can assume their magnitudes to be F. Then, \(\vec{F_A} = F\hat{i}\) (east direction) and \(\vec{F_B} = -F\hat{j}\) (south direction).

Calculating the resultant force

To find the resultant force acting on player B, we need to add the two force vectors \(\vec{F_A}\) and \(\vec{F_B}\). The resultant force \(\vec{R} = \vec{F_A} + \vec{F_B}\) can be calculated as follows: \(\vec{R} = F\hat{i} + (-F\hat{j}) = F\hat{i} - F\hat{j}\)

Finding the direction of the resultant force

To find the direction of the resultant force, we need to calculate the angle \(\theta\) between the resultant force vector and the x-axis (east). We can use the tangent function to calculate the angle: \(\tan{\theta} = \frac{|\vec{R_y}|}{|\vec{R_x}|}\) \(\tan{\theta} = \frac{F}{F}\) \(\tan{\theta} = 1\) The angle \(\theta\) can be calculated as: \(\theta = \tan^{-1}(1)\) \(\theta = 45^\circ\)

Finding the direction of player B's motion

Since the angle between the resultant force and the x-axis (east) is \(45^\circ\), player B is likely to move in the direction that is \(45^\circ\) south of east due to these forces.

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At one moment in a football game, player A exerts a force to the east on player \(\mathrm{B}\). At the same time, a teammate of \(\mathrm{A}\) exerts the samesized force to the south on player \(\mathrm{B}\). In what direction is \(\mathrm{B}\) likely to go because of these forces?

Short Answer

Step by step solution

Representing force vectors

Calculating the resultant force

Finding the direction of the resultant force

Finding the direction of player B's motion

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