A baseball player leads off the game and hits a long home run. The ball leaves the bat at an angle of 30 degrees from the horizontal with a velocity of 40 m/s. How far will it travel in the air?
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To determine how far the baseball will travel in the air, we can use the basic principles of projectile motion. Projectile motion is the motion of an object that is projected into the air and subject only to the force of gravity and air resistance (which we will ignore in this case).
First, we need to break down the initial velocity into its horizontal and vertical components. The vertical component is given by V_y = V * sin(θ), where V is the initial velocity (40 m/s) and θ is the angle with the horizontal (30 degrees). The horizontal component is given by V_x = V * cos(θ).
Since the baseball starts and ends at the same height, the vertical component of its velocity will be zero when it lands. We can calculate the time it takes for the projectile to land by using the equation:
-Δy = V_y * t + (1/2) * g * t^2,
where Δy is the change in height (zero in this case, since the baseball starts and ends at the same height), t is the time of flight, and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Rearranging the equation, we get:
(1/2) * g * t^2 = V_y * t.
Now, we can use the equation for horizontal distance traveled:
Δx = V_x * t.
Since the time of flight is the same for both equations, we can rearrange the equation for horizontal distance to solve for t:
t = Δx / V_x.
Substituting this value of t into the equation for vertical motion, we get:
(1/2) * g * (Δx / V_x)^2 = V_y * (Δx / V_x).
Now, we can rearrange this equation to solve for Δx, which represents the horizontal distance traveled by the baseball:
(1/2) * g * (Δx^2 / V_x^2) = V_y * Δx / V_x.
Simplifying the equation yields:
(1/2) * g * Δx = V_y.
Now we can solve for Δx:
Δx = (2 * V_y) / g.
Plugging in the values, Δx = (2 * 40 * sin(30)) / 9.8.
Calculating this expression, the baseball will travel approximately 87.4 meters in the air.