PLEASE HELP!
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 principles of projectile motion.
Step 1: Find the time of flight:
In projectile motion, the horizontal and vertical motions are independent of each other. The time of flight of the ball can be found using the vertical component of motion.
Since the ball was hit 1.2 m above the ground and landed on the ground, the vertical displacement is given by h = 0 - 1.2 = -1.2 m (taking upward as positive).
Using the equation for vertical displacement in projectile motion: h = (V₀y × t) - (0.5 × g × t²), where V₀y is the initial vertical velocity, g is the acceleration due to gravity (9.8 m/s²), and t is the time of flight.
Plugging in the known values: -1.2 = (V₀y × t) - (0.5 × 9.8 × t²)
Step 2: Find the horizontal distance:
The horizontal distance traveled by the ball can be found using the horizontal component of motion.
Using the equation for horizontal distance in projectile motion: d = V₀x × t, where V₀x is the initial horizontal velocity and t is the time of flight.
The horizontal component of velocity, V₀x, can be found using the initial angle of 39 degrees and the initial speed, which we will assume to be v₀.
V₀x = v₀ × cos(θ), where θ is the angle.
Step 3: Solve for the time of flight:
From Step 1, we have an equation with one unknown value (t). We can solve this equation to find the time of flight.
-1.2 = (V₀y × t) - (0.5 × 9.8 × t²)
Step 4: Solve for the initial speed:
With the time of flight, we can now solve for the initial speed (v₀) using the horizontal distance formula:
d = V₀x × t
Given that the horizontal distance is 80 m, we have:
80 = (v₀ × cos(θ)) × t
Step 5: Calculate the minimum speed:
Now that we have the initial speed (v₀), we can calculate the minimum speed of the ball when it left the bat by finding the resultant velocity (v) of the initial horizontal and vertical velocities.
The resultant velocity is given by:
v = √[(V₀x)² + (V₀y)²]
And since V₀x = v₀ × cos(θ) and V₀y = v₀ × sin(θ):
v = √[(v₀ × cos(θ))² + (v₀ × sin(θ))²]
Simplifying and plugging in the known values, we can calculate the minimum speed of the ball.