Question: A ball is thrown vertically upward with an initial velocity of 15 m/s from a height of 2 meters above the ground. Ignoring air resistance, determine the following:
a) The time it takes for the ball to reach its maximum height. b) The maximum height reached by the ball. c) The total time the ball is in the air. d) The velocity of the ball when it reaches the ground.
Use g = 9.8 m/s^2 for acceleration due to gravity. Round your answers to two decimal places.
Answer:
a) To find the time it takes for the ball to reach its maximum height, we can use the equation of motion:
v = u + at
Where: v = final velocity (0 m/s at the maximum height) u = initial velocity (15 m/s) a = acceleration due to gravity (-9.8 m/s^2)
Rearranging the equation, we have:
t = (v - u) / a
Substituting the values:
t = (0 - 15) / (-9.8)
t = -15 / -9.8
t ≈ 1.53 s
Therefore, it takes approximately 1.53 seconds for the ball to reach its maximum height.
b) To find the maximum height reached by the ball, we can use the equation of motion:
s = ut + (1/2)at²
Where: s = displacement (maximum height) u = initial velocity (15 m/s) t = time taken (1.53 s) a = acceleration due to gravity (-9.8 m/s^2)
Substituting the values:
s = (15 * 1.53) + (1/2)(-9.8)(1.53)²
s = 22.95 + (-11.97)
s ≈ 10.98 m
Therefore, the maximum height reached by the ball is approximately 10.98 meters.
c) The total time the ball is in the air can be calculated by doubling the time it takes to reach the maximum height:
Total time = 2t
Total time = 2 * 1.53
Total time ≈ 3.06 s
Therefore, the total time the ball is in the air is approximately 3.06 seconds.
d) To find the velocity of the ball when it reaches the ground, we can use the equation of motion:
v = u + at
Where: v = final velocity u = initial velocity (15 m/s) a = acceleration due to gravity (-9.8 m/s^2) t = time taken to reach the ground (which is equal to the total time in the air, 3.06 s)
Substituting the values:
v = 15 + (-9.8)(3.06)
v = 15 - 29.988
v ≈ -14.99 m/s
Therefore, the velocity of the ball when it reaches the ground is approximately -14.99 m/s. The negative sign indicates that the ball is moving downward.