Anthony Rizzo hits a baseball in such a way that it leaves his bat traveling at 40.4 m/s at an angle of 26 degrees above the horizontal. When hit, the baseball is 1.2 m above the ground, is in fair territory, and is 127 meters away from a 3.5 m tall fence? Is this a home run?

Okay, thank you. That makes much more sense :)

To determine if this hit is a home run, we need to calculate the maximum distance the baseball will travel and compare it to the distance from the fence.

Step 1: Calculate the vertical component of the initial velocity.
The vertical component of the initial velocity can be calculated using the formula:
v_y = v * sin(theta)

v_y = 40.4 m/s * sin(26°)
v_y ≈ 40.4 m/s * 0.438

v_y ≈ 17.6992 m/s

Step 2: Calculate the time it takes for the baseball to reach its maximum height.
The formula for the time to reach maximum height for a projectile is:
t = v_y / g

Where g is the acceleration due to gravity (approximately 9.8 m/s^2).

t = 17.6992 m/s / 9.8 m/s^2
t ≈ 1.805 s

Step 3: Calculate the maximum height reached by the baseball.
The formula for the maximum height reached by a projectile is:
h_max = (v_y^2) / (2 * g)

h_max = (17.6992 m/s)^2 / (2 * 9.8 m/s^2)
h_max ≈ 157.4132 m

Step 4: Calculate the horizontal component of the displacement.
The horizontal component of the initial velocity can be calculated using the formula:
v_x = v * cos(theta)

v_x = 40.4 m/s * cos(26°)
v_x ≈ 40.4 m/s * 0.897

v_x ≈ 36.2808 m/s

Step 5: Calculate the time it takes for the baseball to reach the fence.
The time it takes for the baseball to reach the fence can be calculated using the formula:
t_total = 2 * t

t_total = 2 * 1.805 s
t_total ≈ 3.61 s

Step 6: Calculate the horizontal distance traveled by the baseball.
The horizontal distance traveled can be calculated using the formula:
d_x = v_x * t_total

d_x = 36.2808 m/s * 3.61 s
d_x ≈ 130.8167 m

Step 7: Compare the horizontal distance to the distance from the fence.
If the horizontal distance traveled exceeds the distance from the fence, it is a home run. Otherwise, it is not a home run.

If the baseball is 127 meters away from the fence and the horizontal distance traveled is approximately 130.8167 meters, then the hit is a home run.

To determine whether it is a home run or not, we need to find out if the baseball will clear the 3.5 m tall fence that is 127 meters away.

First, let's break down the initial velocity of the baseball into its horizontal and vertical components.

The horizontal component of velocity (Vx) remains constant throughout the flight because there is no horizontal acceleration. Therefore, the initial velocity in the horizontal direction is the same as the final velocity.

Vx = V * cos(theta)

where V is the magnitude of the initial velocity (40.4 m/s) and theta is the angle above the horizontal (26 degrees).

Vx = 40.4 m/s * cos(26 degrees)
Vx = 36.437 m/s

Now, let's calculate the time (t) it takes for the baseball to reach the fence 127 meters away. Since there is no vertical acceleration, we can use the horizontal velocity to find the time of flight.

t = distance / Vx

t = 127 m / 36.437 m/s
t ≈ 3.48 s

After 3.48 seconds, the baseball will reach the fence. Now, let's calculate how high the baseball will be at that time.

The vertical component of velocity (Vy) changes due to the effect of gravity. We can use the kinematic equation to find the vertical displacement.

Vy = V * sin(theta)

Vy = 40.4 m/s * sin(26 degrees)
Vy = 17.212 m/s

Using the following kinematic equation:

displacement = initial velocity * time + 0.5 * acceleration * time^2

Since the initial vertical velocity is upwards, acceleration due to gravity is negative (-9.8 m/s^2), and the displacement at the top is zero, we can rearrange the equation to solve for the maximum height (H).

0 = 17.212 m/s * t - 0.5 * 9.8 m/s^2 * t^2

Simplifying, we get:

H = 0.5 * 9.8 m/s^2 * t^2 / 17.212 m/s

H ≈ 6.14 m

Therefore, at the time the baseball reaches the fence, it will be approximately 6.14 m above the ground.

Since the baseball is 3.5 m tall and it will clear the fence by a height of approximately 6.14 m, we can conclude that it is indeed a home run.

just plug in your numbers for the formula for a trajectory. The height

y = 1.2 + (tan26°)x - 4.9/(40.4 cos26°)^2 x^2

is y(127) >= 3.5?