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 the given problem, we will break it down into subproblems:
(a) To determine the impulse acting on the ball, we can use the formula:
Impulse = Change in momentum = Final momentum - Initial momentum
1. Convert the mass of the ball from grams to kilograms:
140 g = 0.140 kg
2. Calculate the initial momentum (p_initial) of the ball:
p_initial = mass × initial velocity
= 0.140 kg × 90.0 mph
3. Convert the initial velocity from mph to m/s:
1 mph = 0.44704 m/s
initial velocity = 90.0 mph × 0.44704 m/s
4. Calculate the final momentum (p_final) of the ball after being batted:
p_final = mass × final velocity
= 0.140 kg × 90.0 mph × 0.44704 m/s
5. Calculate the impulse (I) acting on the ball:
Impulse = p_final - p_initial
(b) To determine the average force on the ball, we can use the formula:
Average Force = Impulse / Impact Time
Given:
Impact time (Δt) = 0.00080 s
Calculate:
Average Force = Impulse / Δt
(c) To find the maximum force (F_max) using the given function F(t) = F_max sin^2(4000t), we need to find the value of F_max that will generate the same impulse as the average force. Since impulse is given by the area under the force-time graph, the impulse can also be calculated by integrating the force function over the time interval (0, Δt) and setting it equal to the average force.
Integrating F(t) with respect to time over the interval (0, Δt) with the given function, we can solve for F_max.
(d) To find the average acceleration of the ball, we can use the formula:
Average Acceleration = Change in Velocity / Impact Time
Given:
Initial velocity = 90.0 mph
Final velocity = 90.0 mph at a 30.0° angle
Calculate:
Change in Velocity = Final velocity - Initial velocity
(e) To determine if the ball clears the 7-foot tall outfield fence, we need to calculate the horizontal distance traveled by the ball. We can use the equation of motion in the x-direction to find the time of flight and then use this time to calculate the horizontal distance traveled.
Given:
Initial velocity = 90.0 mph at a 30.0° angle
Distance to the fence = 410 ft
Height of the fence = 7 ft
Calculate:
1. Calculate the vertical component of initial velocity:
Vertical velocity = Initial velocity × sin(angle)
2. Calculate the time of flight (T):
T = (2 × vertical velocity) / g (where g is the acceleration due to gravity)
3. Calculate the horizontal component of initial velocity:
Horizontal velocity = Initial velocity × cos(angle)
4. Calculate the horizontal distance traveled (D):
D = Horizontal velocity × T
Now we can go ahead and solve each part of the problem using the above equations and given values.