A pitcher throws a baseball horizontally from the mound to home plate. The ball falls 0.926 m (3.04 ft) by the time it reaches home plate 18.3 m (60 ft) away. How fast was the pitcher's pitch?
.926 = (1/2)(9.8) t^2
solve for t
u = 18.3/t
4.29
To determine the speed of the pitcher's pitch, we can start by using the kinematic equation:
\(d = v_0 \cdot t + \frac{1}{2} \cdot a \cdot t^2)
In this case, the ball is thrown horizontally, meaning there is no vertical acceleration (a = 0). Therefore, the equation simplifies to:
\(d = v_0 \cdot t\)
where:
- d is the horizontal distance traveled by the ball (18.3 m)
- v_0 is the initial velocity (pitcher's pitch speed)
- t is the time taken for the ball to reach home plate
We can see that the vertical fall (0.926 m) does not influence the horizontal motion, so it is not needed in this calculation.
We rearrange the equation to solve for v_0:
\(v_0 = \frac{d}{t}\)
Substituting the given values:
\(v_0 = \frac{18.3 m}{t}\)
Please provide the time taken for the ball to reach home plate so that we can calculate the pitch speed accurately.
To find the speed of the pitcher's pitch, we need to use the equation of motion for projectile motion.
The equation is:
d = v * t
where:
d is the distance traveled by the ball (in this case, 18.3 m),
v is the velocity or speed of the ball (what we're trying to find), and
t is the time taken for the ball to travel the distance.
Since the pitcher threw the ball horizontally, the initial vertical velocity (upward) is zero, and the only motion to consider is horizontal. Hence, the time taken is the same for both the horizontal and vertical components.
To find the time taken, we can use the equation:
d = 0.5 * g * t^2
where:
d is the vertical displacement (in this case, 0.926 m),
g is the acceleration due to gravity (9.8 m/s^2 approx.), and
t is the time taken.
Rearranging the equation, we get:
t = sqrt(2 * d / g)
Substituting the values:
t = sqrt(2 * 0.926 m / 9.8 m/s^2)
t = sqrt(0.1884)
t ≈ 0.434 seconds (rounded to three decimal places)
Now that we have the time taken, we can use the initial equation to find the speed:
v = d / t
v = 18.3 m / 0.434 s
v ≈ 42.2 m/s (rounded to three decimal places)
Therefore, the speed of the pitcher's pitch was approximately 42.2 m/s.