A 140 g, 90.0 mph fastball comes across the plate very nearly horizontally and is batted at
90.0 mph at a 30.0O angle (above the horizontal) toward center field. (a) Determine the impulse
(in N.s) acting on the ball. (b) If the impact time is 0.00080 s, determine the average force on the
ball. (c) The force on the ball as a function of time can be modeled as F(t) = Fmax sin2(4000t).
Using this function, assuming it will give the same impulse as the average force, find Fmax. (d)
Find the average acceleration of the ball. (e) For some extra 2-D motion practice, does this ball
clear the 7 foot tall outfield fence, 410 ft away?
To solve this problem, we'll need to use principles of physics, specifically related to impulse, force, acceleration, and projectile motion.
(a) To determine the impulse acting on the ball, we can use the formula:
Impulse = change in momentum
Momentum before = mass * initial velocity = (140 g) * (90.0 mph)
Momentum after = mass * final velocity = (140 g) * (90.0 mph) * sin(30°)
Impulse = Momentum after - Momentum before
To calculate this, we need to convert mass from grams to kilograms and velocity from mph to m/s:
Mass = 140 g = 0.14 kg
Initial velocity = 90.0 mph = 40.23 m/s (approx.)
Final velocity = 90.0 mph = 40.23 m/s (approx.) * sin(30°)
Substituting the values, we can find the impulse.
(b) To determine the average force on the ball, we can use the formula:
Force = Impulse / Time
In this case, the Time is given as 0.00080 s.
(c) To find Fmax using the given function F(t) = Fmax sin²(4000t), we need to equate impulse with the average force. Recall that impulse is equal to the area under the force-time curve. So, integrating F(t) over time will give us the impulse.
Impulse = ∫[0 to t] Fmax sin²(4000t) dt
Solving this integral will give us the value of Fmax.
(d) Average acceleration can be calculated using the formula:
Average acceleration = Change in velocity / Time taken
Here, the change in velocity is the difference between the initial and final velocity. We can use the x and y components of velocity separately for this calculation, considering the projectile motion.
(e) To calculate if the ball clears the 7-foot tall outfield fence, we need to analyze the ball's trajectory. We'll need to find the maximum height reached by the ball and see if it surpasses 7 feet. And we also need to calculate the horizontal distance traveled by the ball and check if it exceeds 410 feet.
By combining the equations of projectile motion and using the known initial velocity, angle, and acceleration due to gravity, we can find the necessary information to determine if the ball clears the fence.
These calculations involve a number of steps and calculations which would be best done using a calculator or a programming language.