# A football is thrown upward at an 48◦

angle
to the horizontal.
The acceleration of gravity is 9.8 m/s^2.
To throw an 80.4 m pass, what must be
the initial speed of the ball?

## the range of a projectile is

... (launch vel)^2 * sin(2 * launch angle) / g

80.4 = v^2 * sin(2 * 48º) / 9.8

## Well, to throw a pass, you need a good arm and a powerful throw! So let's calculate the initial speed, shall we?

Since the football is thrown upwards at an angle of 48 degrees to the horizontal, we can break it down into its horizontal and vertical components. The horizontal component doesn't affect the distance traveled, so we'll focus on the vertical component.

The formula we can use is:

d = v^2 / (2g)

where d is the distance, v is the initial velocity, and g is the acceleration due to gravity.

In this case, the distance is 80.4 m and gravity is 9.8 m/s^2.

Plugging these values into the formula, we get:

80.4 = v^2 / (2 * 9.8)

Now, let's solve for v:

v^2 = 80.4 * 2 * 9.8
v^2 = 1576.8
v ≈ √1576.8
v ≈ 39.7 m/s

Therefore, the initial speed of the ball should be approximately 39.7 m/s.

Now, go out there and throw that pass like a pro! Just don't throw it at me, I'm a clown bot, not a football bot!

## To find the initial speed of the ball, we can break down the motion into the horizontal and vertical components.

- Angle of projection (θ) = 48 degrees
- Distance traveled in the horizontal direction (range) = 80.4 meters
- Acceleration due to gravity (g) = 9.8 m/s^2

2. Determine the horizontal component of the initial velocity (V₀x) using the formula:
V₀x = V₀ * cos(θ)
where V₀x is the initial horizontal velocity and V₀ is the initial speed.

3. Calculate the vertical component of the initial velocity (V₀y) using the formula:
V₀y = V₀ * sin(θ)
where V₀y is the initial vertical velocity.

4. The time taken for the ball to reach the highest point can be calculated using the formula:
t = V₀y / g
where t is the time taken.

5. Since the total time of flight is twice the time taken to reach the highest point, we can calculate the total time (T) using the formula:
T = 2 * t

6. Using the formula for the range of projectile motion:
range = V₀x * T
substitute the values of range and T from steps 2 and 5.

7. Rearrange the formula to solve for V₀:
V₀ = range / (2 * t * cos(θ))

8. Substitute the values of range, t, and θ from steps 2, 4, and 1 to calculate V₀.

9. Calculate V₀ to find the initial speed of the ball.

Following these steps, we can find the initial speed of the ball.

## To solve this problem, we can break the initial velocity into its horizontal and vertical components.

Given:
Angle of projection (θ) = 48°
Gravitational acceleration (g) = 9.8 m/s²
Range of the pass (R) = 80.4 m

First, let's calculate the vertical component of the initial velocity (v₀y):
v₀y = v₀ * sin(θ)

Next, we can find the time taken for the ball to reach its maximum height:
The ball reaches its maximum height when its vertical velocity (v_y) becomes zero. We can use the following equation to find the time taken (t) to reach maximum height:
v_y = v₀y - g * t

Since v_y = 0 at the maximum height, we have:
0 = v₀ * sin(θ) - g * t

Solving this equation for t, we get:
t = v₀ * sin(θ) / g

The time taken to reach the maximum height is equal to the total time of flight (T), so:
T = 2 * t

Finally, we can use the formula for the range of projectile motion to find the initial speed (v₀) of the ball:
R = v₀ * cos(θ) * T

Substituting the values we know, the equation becomes:
80.4 = v₀ * cos(48°) * 2 * (v₀ * sin(48°) / 9.8)

Simplifying:
80.4 = 2 * v₀² * cos(48°) * sin(48°) / 9.8

We can solve this equation to find the value of v₀.