a tennis ball is thrown straight up with an initial velocity of 22.5 m/s. it is caught at the same distance above the ground.
a). how long does it take to reach its maximum height?
b). how high does the ball rise?
c). at what speed does it hit the ground?
d). what total length of time is the ball in the air
a. V = Vo + gt
t = -Vo/g = -22.5/-9.8 = 2.3 s.
b. b. V^2 = Vo^2 + 2g*h
h = (V^2-Vo^2)/2g
h = (0-(22,5)^2)/-19.6 = 25.83 m.
c. V^2 = Vo^2 + 2g*h
V^2 = 0 + 19.6*25.83 = 506.27
V = 22.5 m/s.
d. T = Tr + Tf = 2.3 + 2.3 = 4.6 s.
How long will it take the ball to reach the peak of the trajectory?
How high does the ball rise?
How high does the ball rise
a) Well, you know what they say, "What goes up must come down." It takes approximately half of the total time in the air for the tennis ball to reach its maximum height. So, let's do some quick calculations...carry the one...ah yes! It takes 11.25 seconds to reach its maximum height.
b) Ah, the ball's high aspirations! The distance it rises can be calculated using the good old kinematic equations. With an initial velocity of 22.5 m/s and a time of 11.25 seconds to reach its peak, the ball rises to a height of approximately 126.6 meters above the ground. That's quite the leap!
c) A hard-hitting question indeed! To determine the speed at which the ball hits the ground, we need to consider its downward journey. Using lovely physics again, we calculate the final velocity when it hits the ground to be approximately 50.4 m/s. Ouch, that's quite a smashing speed!
d) Ah, the total time in the air. To determine this, we need to account for the time it takes to reach the maximum height and the time it takes to come back down. Adding these two together, we find that the ball is in the air for approximately 22.5 seconds. That's plenty of airtime for a bouncy ball like this one!
To solve these questions, we can use the equations of motion for an object in freefall.
a) To find the time it takes for the tennis ball to reach its maximum height, we can use the fact that at the maximum height, the ball's final velocity is 0 m/s. We can use the equation:
v = u + gt
where:
v = final velocity (0 m/s at maximum height)
u = initial velocity (22.5 m/s)
g = acceleration due to gravity (-9.8 m/s^2)
Rearranging the equation to solve for time (t), we get:
t = (v - u) / g
Substituting the given values, we find:
t = (0 - 22.5) / (-9.8) = 2.30 seconds
So, it takes approximately 2.30 seconds for the tennis ball to reach its maximum height.
b) To find the height the ball rises, we can use the equation:
s = ut + (1/2)gt^2
where:
s = height (to be determined)
u = initial velocity (22.5 m/s)
t = time taken to reach maximum height (2.30 seconds)
g = acceleration due to gravity (-9.8 m/s^2)
Substituting the given values, we have:
s = (22.5)(2.30) + (1/2)(-9.8)(2.30)^2
Simplifying the equation:
s = 25.88 meters
Therefore, the ball rises to a height of approximately 25.88 meters.
c) To find the speed at which the ball hits the ground, we can use the equation:
v = u + gt
where:
v = final velocity (to be determined)
u = initial velocity (22.5 m/s)
g = acceleration due to gravity (-9.8 m/s^2)
At the ground, the ball's final velocity is the same as its initial velocity, but negative since it's moving in the opposite direction. Therefore, we have:
v = -u
Substituting the given value, we find:
v = -22.5 m/s
So, the ball hits the ground with a speed of 22.5 m/s.
d) To find the total length of time the ball is in the air, we double the time it takes to reach the maximum height.
Total time = 2t = 2(2.30) = 4.60 seconds
Thus, the ball is in the air for a total of approximately 4.60 seconds.