a bowler releases her bowling ball at 425.36 cm/s.it decelerates towards the pins at -4.8 cm/s^2.the length of a bowling alley is 60ft.what velocity did the ball hit the pins and how long did it take to get there?
To find the velocity and time it takes for the bowling ball to hit the pins, we can use the equations of motion.
First, let's convert the given units to a consistent system. We will convert the speed from cm/s to ft/s and the acceleration from cm/s^2 to ft/s^2.
1 ft = 30.48 cm
Given:
Initial speed (u) = 425.36 cm/s = (425.36/30.48) ft/s ≈ 13.96 ft/s
Acceleration (a) = -4.8 cm/s^2 = (-4.8/30.48) ft/s^2 ≈ -0.157 ft/s^2
Distance (s) = Length of the bowling alley = 60 ft
Using the second equation of motion:
v^2 = u^2 + 2as
We can rearrange this equation to solve for the final velocity (v):
v^2 = u^2 + 2as
Since the acceleration (a) is negative (deceleration), we will consider the final velocity as negative as well.
v^2 = (13.96 ft/s)^2 + 2(-0.157 ft/s^2)(60 ft)
v^2 = 194.8816 + (-18.84)
v^2 ≈ 176.0416
Taking the square root of both sides:
v ≈ -13.27 ft/s
Therefore, the ball hit the pins with a velocity of approximately -13.27 ft/s. (Note: The negative sign indicates that the ball is moving in the opposite direction of the initial velocity).
To find the time it took to get there, we can use the first equation of motion:
v = u + at
Substituting the given values:
-13.27 ft/s = 13.96 ft/s + (-0.157 ft/s^2) * t
Simplifying:
-13.27 ft/s - 13.96 ft/s = -0.157 ft/s^2 * t
-27.23 ft/s = -0.157 ft/s^2 * t
Dividing both sides by -0.157 ft/s^2:
t ≈ 173.6 seconds
Therefore, it took approximately 173.6 seconds for the ball to hit the pins.