A baseball player hits a home run over the left-field fence, which is 104 m from home plate. The ball is hit at a point 1.04 m directly above home plate, with an initial velocity directed 28.5° above the horizontal. By what distance does the baseball clear the 3.00 m high fence, if it passes over it 2.80 s after being hit?
The ball's horizontal velocity component is a constant 104/2.80 = 37.14 m/s. The initial vertical velocity component must be 37.14 tan28.5 = 20.17 m/s
Next, compute the height Y of the ball 2.80 s after it is hit
Y = 1.04 + 20.17*2.80 - (g/2)*(2.80)^2
Subtract 3.0 m from that for the fence clearance distance.
lets go boss
To find the distance the baseball clears the fence, we first need to determine the horizontal distance the ball traveled in 2.80 seconds.
Step 1: Determine the horizontal component of the initial velocity.
The horizontal component of the initial velocity can be found using the formula:
Vx = V * cos(theta)
where V is the magnitude of the initial velocity and theta is the angle above the horizontal.
Given:
V = initial velocity = ?
theta = angle above the horizontal = 28.5°
Using the given values, we can now calculate the horizontal component of the initial velocity:
Vx = V * cos(theta)
Vx = V * cos(28.5°)
Step 2: Determine the horizontal distance traveled by the baseball in 2.80 seconds.
The horizontal distance traveled can be calculated using the formula:
x = Vx * t
where t is the time elapsed.
Given:
t = 2.80 s
Using the given values and the horizontal component of the initial velocity from Step 1, we can now calculate the horizontal distance traveled by the baseball:
x = Vx * t
Step 3: Calculate the vertical distance traveled by the baseball in 2.80 seconds.
The vertical distance traveled can be calculated using the formula:
y = Vy * t + (1/2) * g * t^2
where Vy is the vertical component of the initial velocity and g is the acceleration due to gravity.
Given:
t = 2.80 s
g = acceleration due to gravity = 9.8 m/s^2
Using the given values and the initial velocity from Step 1, we can now calculate the vertical distance traveled by the baseball:
y = Vy * t + (1/2) * g * t^2
Step 4: Determine the total distance the baseball cleared the fence.
The total distance the baseball cleared the fence can be calculated using the formula:
d = sqrt(x^2 + y^2) - H
where x is the horizontal distance traveled, y is the vertical distance traveled, and H is the height of the fence.
Given:
H = height of the fence = 3.00 m
Using the values calculated in Step 2 and Step 3, we can now calculate the total distance the baseball cleared the fence:
d = sqrt(x^2 + y^2) - H
Following the four steps listed above, you can input the values and calculate the distance the baseball clears the 3.00 m high fence.
To find the distance by which the baseball clears the fence, we can break the problem into two parts: the horizontal distance traveled by the ball and the vertical distance the ball clears the fence.
First, let's find the horizontal distance traveled by the ball. We know that the initial velocity of the ball is 28.5° above the horizontal. We can split this initial velocity into its horizontal and vertical components.
The horizontal component is given by: Vx = V * cos(θ)
Vx = V * cos(28.5°)
where V is the initial velocity of the ball.
The vertical component is given by: Vy = V * sin(θ)
Vy = V * sin(28.5°)
where V is the initial velocity of the ball.
Next, we can use the horizontal component of the velocity to find the horizontal distance traveled by the ball. Since the time it takes for the ball to clear the fence is given (2.80 s), we can use the formula:
d = Vx * t
where d is the distance traveled, Vx is the horizontal component of the velocity, and t is the time it takes for the ball to clear the fence.
Now, let's find the vertical distance the ball clears the fence. We know that the ball passes over the fence 2.80 s after being hit, so we can use the equation of motion for vertical motion:
y = y0 + (Vy * t) - (0.5 * g * t^2)
where y is the vertical position of the ball, y0 is the initial vertical position of the ball, Vy is the vertical component of the velocity, t is the time it takes for the ball to clear the fence, and g is the acceleration due to gravity.
The initial vertical position of the ball is given as 1.04 m above the home plate.
Finally, to find the distance by which the baseball clears the fence, we can subtract the height of the fence from the vertical distance the ball clears the fence:
clearance = vertical distance - height of the fence.
Plug in the given values and calculate each part of the problem step by step to find the clearance.