To solve this, we can apply the principles of projectile motion. Given dataTo solve this, we can apply the principles of projectile motion. Given

• Launch angle, $\theta = 37.0^\circ$
• Initial velocity, $v_0 = 25.0 \, \text{m/s}$

We want to find the horizontal range of the football, or how far it travels horizontally. This requires calculating both the horizontal and vertical components of the initial velocity, determining the time of flight, and then finding the range.

Step-by-Step Solution:

1. Resolve the Initial Velocity into Components:

   • Horizontal component, $v_{0x} = v_0 \cos \theta$
   • Vertical component, $v_{0y} = v_0 \sin \theta$

   Using $\theta = 37.0^\circ$: $v_{0x} = 25.0 \cos 37^\circ$ $v_{0y} = 25.0 \sin 37^\circ$

2. Calculate the Time of Flight: The time of flight $T$ can be found using the formula: $T = \frac{2 v_{0y}}{g}$ where $g = 9.8 \, \text{m/s}^2$ is the acceleration due to gravity.

3. Calculate the Horizontal Range: The horizontal range $R$ is given by: $R = v_{0x} \cdot T$

Solution:

Let's go through these calculations:

1. Calculate $v_{0x}$ and $v_{0y}$:

   • $v_{0x} = 25.0 \cos 37^\circ \approx 25.0 \times 0.7986 = 19.965 \, \text{m/s}$
   • $v_{0y} = 25.0 \sin 37^\circ \approx 25.0 \times 0.6018 = 15.045 \, \text{m/s}$

2. Time of Flight $T$: $T = \frac{2 \times 15.045}{9.8} \approx 3.07 \, \text{seconds}$

3. Horizontal Range $R$: $R = v_{0x} \cdot T = 19.965 \times 3.07 \approx 61.3 \, \text{meters}$

Answer:

The football travels approximately 61.3 meters horizontally.

Would you like more details on any of these steps or have any questions?

Related Questions:

1. What would the horizontal range be if the initial velocity were 30 m/s?
2. How does changing the launch angle affect the range of the projectile?
3. How long does the football stay in the air (time of flight) if launched at a different angle?
4. What is the maximum height the football reaches during its trajectory?
5. How would air resistance alter the football's range?

Tip:

For projectile problems, breaking the initial velocity into horizontal and vertical components is essential for calculating both time of flight and range.