Answer:
The baseball's height above the ground is approximately 21.4 meters.
Step-by-step explanation:
To analyze the motion of the baseball, we can use the laws of physics, specifically the equations of motion under constant acceleration due to gravity. The acceleration due to gravity on Earth is approximately 9.8 m/s², and we'll consider it to be negative since it acts in the opposite direction of the motion (downward).
Let's break down the problem step by step:
Initial velocity (u): 14 m/s (upward, but we'll consider it positive)
Initial height (h): 13 m
Acceleration due to gravity (g): -9.8 m/s² (negative because it acts downward)
Final velocity (v): ?
Time taken to reach the ground (t): ?
Height above the ground at a given time (y): ?
Step 1: Calculate the time taken to reach the ground (t):
When the baseball reaches the ground, its height (y) will be 0. We can use the following equation to find the time (t) it takes to reach the ground:
y = h + ut + (1/2)gt²
Where:
y = final height (0 m, since it reaches the ground)
h = initial height (13 m)
u = initial velocity (14 m/s)
g = acceleration due to gravity (-9.8 m/s²)
t = time taken to reach the ground (unknown)
0 = 13 + 14t + (1/2)(-9.8)t²
Now, we need to solve this quadratic equation for t. The equation can be written as:
-4.9t² + 14t + 13 = 0
Step 2: Solve for t using the quadratic formula:
t = (-b ± √(b² - 4ac)) / 2a
where a = -4.9, b = 14, and c = 13.
t = [ -14 ± √(14² - 4(-4.9)(13)) ] / 2(-4.9)
t = [ -14 ± √(196 + 254.8) ] / -9.8
t = [ -14 ± √450.8 ] / -9.8
t = [ -14 ± 21.24 ] / -9.8
Now we'll get two values for t (due to the ± symbol):
t₁ = ( -14 + 21.24 ) / -9.8 ≈ -0.73 seconds (unrealistic, so we'll discard this)
t₂ = ( -14 - 21.24 ) / -9.8 ≈ 3.05 seconds
Step 3: Calculate the final velocity (v):
We can use the following equation to find the final velocity (v) at the time it reaches the ground:
v = u + gt
where:
v = final velocity (unknown)
u = initial velocity (14 m/s)
g = acceleration due to gravity (-9.8 m/s²)
t = time taken to reach the ground (3.05 seconds)
v = 14 + (-9.8)(3.05)
v ≈ 14 - 29.89
v ≈ -15.89 m/s
Step 4: Calculate the height above the ground at a given time:
At any given time (t) during the motion, we can calculate the height above the ground (y) using the equation:
y = h + ut + (1/2)gt²
where:
y = height above the ground (unknown)
h = initial height (13 m)
u = initial velocity (14 m/s)
g = acceleration due to gravity (-9.8 m/s²)
t = given time
For example, let's find the height at t = 2 seconds:
y = 13 + 14(2) + (1/2)(-9.8)(2)²
y = 13 + 28 + (-19.6)
y ≈ 21.4 meters
So, at t = 2 seconds, the baseball's height above the ground is approximately 21.4 meters.