When Babe Ruth hit a homer over the 10 m -high right-field fence 80 m from home plate, roughly what was the minimum speed of the ball when it left the bat? Assume the ball was hit 1.2 m above the ground and its path initially made a 39 ∘ angle with the ground.
To find the minimum speed of the ball when it left the bat, we can use the following steps:
Step 1: Break down the initial velocity into horizontal and vertical components.
The horizontal component (Vx) remains constant throughout the motion, while the vertical component (Vy) changes due to the effect of gravity.
Given:
Initial angle (θ) = 39°
Initial velocity (V) = ?
Takeoff height (y) = 1.2 m
Acceleration due to gravity (g) = 9.8 m/s²
Using trigonometry, we can find the initial vertical and horizontal velocities as follows:
Vy = V * sin(θ)
Vx = V * cos(θ)
Step 2: Find the time of flight (t).
The time of flight can be calculated using the formula:
t = (2 * Vy) / g
Step 3: Find the total horizontal distance (d).
The total horizontal distance can be calculated using the formula:
d = Vx * t
Step 4: Find the vertical distance covered (h).
The vertical distance covered is the difference between the takeoff height and the fence height:
h = 10 m - y
Step 5: Use the range equation to calculate the initial velocity (V).
The range equation is:
d = Vx * t
Now, we can plug in the values and calculate the minimum speed of the ball when it left the bat.
Note: Since the question asks for the minimum speed, we can assume there is no air resistance.
Let's calculate:
Step 1:
Vy = V * sin(θ)
Vx = V * cos(θ)
Step 2:
t = (2 * Vy) / g
Step 3:
d = Vx * t
Step 4:
h = 10 m - y
Step 5:
d = Vx * t
Using the given values, we can calculate each step.
To find the minimum speed of the ball when it left the bat, we can use the principles of projectile motion.
Here's how you can solve the problem step by step:
Step 1: Break down the initial velocity of the ball into its horizontal and vertical components. The initial velocity can be expressed as V₀ = V₀x + V₀y, where V₀x is the horizontal component and V₀y is the vertical component.
Step 2: Use the given angle of 39° and the initial velocity components to calculate the values of V₀x and V₀y. V₀x can be calculated as V₀x = V₀ * cos(θ), and V₀y can be calculated as V₀y = V₀ * sin(θ), where θ is the given angle.
Step 3: Determine the time it takes for the ball to reach the height of the fence. The vertical component of the velocity (V₀y) can be used to calculate the time using the equation: V₀y = gt - (1/2)g(t²), where g is the acceleration due to gravity (approximately 9.8 m/s²).
Step 4: Find the horizontal distance the ball travels during that time. Use the horizontal component of the velocity (V₀x) and the time calculated in Step 3 to compute the horizontal distance using the equation: D = V₀x * t, where D represents the horizontal distance.
Step 5: Compare the value of the horizontal distance obtained in Step 4 with the given distance of 80 meters. If the horizontal distance is equal to or greater than 80 meters, then the minimum speed has been reached. If not, increase the initial speed of the ball and repeat the calculations until the condition is met.
By iterating this process, you can find the minimum speed required for the ball to clear the fence.