A baseball is hit at 35.0m/s at an angle of 25.0 degrees to the horizontal.

How high will the ball go>
When will it reach that height?
How far across the baseball field will the ball travel?

Thanks.

How high will the ball go>

When will it reach that height?
How far across the baseball field will the ball travel?

The height reached is derivable from h = V^2(sin^2(µ)/2g where V = the launch velocity, µ = the launch angle and g = the acceleration due to gravity.

Using the verical component of the launch velocity, the time to reach the maximum height derives from Vf = Vo - gt where Vf = the final vertical velocity = 0, Vo = the initial vertical velocity = 25sin25º and g is as defined earlier.

The horizontal distance traveled derives from d = V^2sin(2µ)/2g.

To determine the height, time of flight, and horizontal distance traveled by the baseball, we can use the following equations of projectile motion:

1. Maximum height (h):
The vertical component of the baseball's initial velocity will determine the maximum height the ball reaches. We can use the equation:
h = (v^2 * sin^2θ) / (2 * g),
where v is the initial velocity and θ is the angle of projection, and g is the acceleration due to gravity (9.8 m/s^2).

2. Time of flight (t):
The time it takes for the baseball to reach its maximum height and then return to the same height is given by:
t = (2 * v * sinθ) / g.

3. Horizontal distance traveled (d):
The horizontal distance the baseball travels can be calculated using:
d = v * cosθ * t.

Now let's calculate:

Given:
v = 35.0 m/s (initial velocity),
θ = 25.0 degrees (angle of projection),
g = 9.8 m/s^2 (acceleration due to gravity).

1. Maximum height (h):
Using the formula: h = (v^2 * sin^2θ) / (2 * g),
h = (35.0^2 * sin^2 25.0) / (2 * 9.8)
h ≈ 16.65 meters (rounded to two decimal places).

2. Time of flight (t):
Using the formula: t = (2 * v * sinθ) / g,
t = (2 * 35.0 * sin 25.0) / 9.8
t ≈ 3.67 seconds (rounded to two decimal places).

3. Horizontal distance traveled (d):
Using the formula: d = v * cosθ * t,
d = 35.0 * cos 25.0 * 3.67
d ≈ 97.49 meters (rounded to two decimal places).

So, in summary:
- The ball will reach a maximum height of approximately 16.65 meters.
- It will take approximately 3.67 seconds to reach that height.
- The ball will travel approximately 97.49 meters across the baseball field.

To answer these questions, we can use the equations of projectile motion. Let's break it down step by step.

1. How high will the ball go?
To find the maximum height, we need to determine the vertical component of the initial velocity. We can do this using the formula:

Vy = V * sin(theta)

Vy represents the vertical component of the velocity, V is the initial velocity, and theta is the angle.

Given:
V = 35.0 m/s
theta = 25.0 degrees

Using the formula:

Vy = 35.0 * sin(25.0)

Calculating the value of Vy, we find:
Vy ≈ 14.79 m/s

Next, we can use this value to calculate the time taken to reach maximum height. The time of flight to reach maximum height is given by:

t = Vy / g

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

Using the formula:

t = 14.79 / 9.8

Calculating the value of t, we find:
t ≈ 1.51 s

To find the maximum height, we can use the formula:

H = Vy^2 / (2 * g)

Substituting the known values:

H = (14.79^2) / (2 * 9.8)

Calculating the value of H, we find:
H ≈ 11.10 m

So, the ball will reach a maximum height of approximately 11.10 meters.

2. When will it reach that height?
To determine the time taken to reach the maximum height, we can use the same formula as before:

t = Vy / g

Substituting the known values:

t = 14.79 / 9.8

Calculating the value of t, we find:
t ≈ 1.51 s

Therefore, it will take approximately 1.51 seconds for the ball to reach its maximum height.

3. How far across the baseball field will the ball travel?
To find the horizontal distance traveled by the ball, we can use the total time of flight and the horizontal component of the initial velocity. The horizontal component of the velocity is given by:

Vx = V * cos(theta)

Given:
V = 35.0 m/s
theta = 25.0 degrees

Using the formula:

Vx = 35.0 * cos(25.0)

Calculating the value of Vx, we find:
Vx ≈ 31.52 m/s

Now, we can use the formula for horizontal distance:

D = Vx * t

Substituting the known values:

D = 31.52 * 1.51

Calculating the value of D, we find:
D ≈ 47.59 m

Therefore, the ball will travel approximately 47.59 meters across the baseball field.