A quarterback is standing on the football field preparing to throw a pass. His receiver is standing 24 yd down the field and 16 yd to the quarterback's left. The quarterback throws the ball at a velocity of 58 mph toward the receiver at an upward angle of 31°. Write the initial velocity vector of the ball, 
, in component form.

x = 24 yd
y = 16 yd

We are asked to find the initial velocity vector of the football thrown by the quarterback. The vector must be expressed in component form based on the given information: the receiver's position, the throw's speed, and the angle at which the ball is thrown.

### Step 1: Convert velocity from mph to yd/s
The velocity of the ball is given as 58 mph. We need to convert this to yards per second (yd/s) since the distances are in yards.

1 mile = 1760 yards  
1 hour = 3600 seconds

So, to convert 58 mph to yards per second:
\[
58 \ \text{mph} = 58 \times \frac{1760 \ \text{yd}}{1 \ \text{mile}} \times \frac{1 \ \text{hour}}{3600 \ \text{seconds}} = \frac{58 \times 1760}{3600} \ \text{yd/s}
\]

\[
58 \ \text{mph} \approx 28.37 \ \text{yd/s}
\]

### Step 2: Determine the direction of the throw
The receiver is located 24 yards down the field (along the \(x\)-axis) and 16 yards to the quarterback's left (along the \(y\)-axis). The vector pointing to the receiver can be represented as \((24, 16)\).

The angle \(\theta\) of this vector with respect to the positive \(x\)-axis is:
\[
\theta = \tan^{-1} \left( \frac{16}{24} \right) = \tan^{-1} \left( \frac{2}{3} \right) \approx 33.69^\circ
\]

This gives us the horizontal angle at which the ball is thrown relative to the field.

### Step 3: Break down the velocity into components
The initial velocity of the ball is given at an upward angle of \(31^\circ\). We now express this initial velocity vector in 3D components:

- The \(x\)-axis represents the downfield direction (24 yards).
- The \(y\)-axis represents the lateral (left-right) direction (16 yards).
- The \(z\)-axis represents the vertical direction (up-down).

The velocity components are:
\[
v_x = v \cos(31^\circ) \cos(33.69^\circ)
\]
\[
v_y = v \cos(31^\circ) \sin(33.69^\circ)
\]
\[
v_z = v \sin(31^\circ)
\]

Substitute \(v = 28.37 \ \text{yd/s}\):
\[
v_x = 28.37 \cos(31^\circ) \cos(33.69^\circ)
\]
\[
v_y = 28.37 \cos(31^\circ) \sin(33.69^\circ)
\]
\[
v_z = 28.37 \sin(31^\circ)
\]

### Step 4: Calculate the components
Using trigonometric values:
- \(\cos(31^\circ) \approx 0.857\)
- \(\sin(31^\circ) \approx 0.515\)
- \(\cos(33.69^\circ) \approx 0.832\)
- \(\sin(33.69^\circ) \approx 0.555\)

Now, calculate each component:

\[
v_x \approx 28.37 \times 0.857 \times 0.832 \approx 20.24 \ \text{yd/s}
\]
\[
v_y \approx 28.37 \times 0.857 \times 0.555 \approx 13.50 \ \text{yd/s}
\]
\[
v_z \approx 28.37 \times 0.515 \approx 14.62 \ \text{yd/s}
\]

### Final Answer:
The initial velocity vector of the ball in component form is approximately:
\[
\mathbf{v} = \langle 20.24, 13.50, 14.62 \rangle \ \text{yd/s}
\]

Would you like more details on any part of this explanation? 

Here are 5 related questions to expand your understanding:
1. How would the velocity vector change if the receiver were 30 yards down the field?
2. What is the magnitude of the initial velocity vector?
3. How do the velocity components affect the trajectory of the ball?
4. How does the angle of elevation impact the horizontal distance traveled?
5. What happens if the quarterback throws the ball at a different speed?

**Tip:** When breaking down a vector into components, always ensure the units are consistent to avoid calculation errors.

A quarterback is standing on the football field preparing to throw a pass. His receiver is standing 24 yd down the field and 16 yd to the quarterback's left. The quarterback throws the ball at a velocity of 58 mph toward the receiver at an upward angle of 31°. Write the initial velocity vector of the ball in component form.

This problem involves finding the initial velocity vector of a football thrown by a quarterback, with a given speed of 58 mph at an upward angle of 31°. Using vector decomposition and trigonometry, we break down the velocity into horizontal and vertical components. Learn the step-by-step method, suitable for students in Grades 11-12 or introductory college-level physics.

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