Questions: What is the initial velocity of the ball relative to the quarterback? (Assume downfield to be in the +x-direction. Enter the magnitude in m / s and the direction in degrees counterclockwise from the +x-axis.) magnitude 16.04 m / s direction 31.66 0° (counterclockwise from the +x-axis)

What is the initial velocity of the ball relative to the quarterback? (Assume downfield to be in the +x-direction. Enter the magnitude in m / s and the direction in degrees counterclockwise from the +x-axis.) magnitude 16.04 m / s direction 31.66 0° (counterclockwise from the +x-axis)
Transcript text: What is the initial velocity of the ball relative to the quarterback? (Assume downfield to be in the $+x$-direction. Enter the magnitude in $\mathrm{m} / \mathrm{s}$ and the direction in degrees counterclockwise from the $+x$-axis.) magnitude 16.04 $\times \mathrm{m} / \mathrm{s}$ direction 31.66 $0^{\circ}$ (counterclockwise from the $+x$-axis)
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Solution

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Solution Steps

Step 1: Understanding the Problem

We need to determine the initial velocity of a ball relative to the quarterback. The problem provides the magnitude and direction of the velocity.

Step 2: Given Data
  • Magnitude of the velocity: \( 16.04 \, \mathrm{m/s} \)
  • Direction: \( 31.66^\circ \) counterclockwise from the \( +x \)-axis
Step 3: Interpreting the Direction

The direction is given as \( 31.66^\circ \) counterclockwise from the \( +x \)-axis. This means the velocity vector forms an angle of \( 31.66^\circ \) with the positive \( x \)-axis.

Step 4: Calculating the Components of the Velocity

To find the components of the velocity, we use trigonometric functions:

  • \( v_x = v \cos(\theta) \)
  • \( v_y = v \sin(\theta) \)

where \( v = 16.04 \, \mathrm{m/s} \) and \( \theta = 31.66^\circ \).

Step 5: Computing the \( x \)-Component

\[ v_x = 16.04 \cos(31.66^\circ) \] Using a calculator: \[ v_x \approx 16.04 \times 0.8526 \] \[ v_x \approx 13.68 \, \mathrm{m/s} \]

Step 6: Computing the \( y \)-Component

\[ v_y = 16.04 \sin(31.66^\circ) \] Using a calculator: \[ v_y \approx 16.04 \times 0.5261 \] \[ v_y \approx 8.44 \, \mathrm{m/s} \]

Final Answer

The initial velocity of the ball relative to the quarterback is:

  • Magnitude: \( \boxed{16.04 \, \mathrm{m/s}} \)
  • Direction: \( \boxed{31.66^\circ} \) counterclockwise from the \( +x \)-axis
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