a)
After 2 seconds:
Vertical velocity = initial vertical velocity - (acceleration due to gravity x time)
Vertical velocity = 25 m/s - (9.81 m/s^2 x 2 s) = 25 m/s - 19.62 m/s = 5.38 m/s
Vertical position = initial vertical position + (initial vertical velocity x time) - (0.5 x acceleration due to gravity x time^2)
Vertical position = 0 + (25 m/s x 2 s) - (0.5 x 9.81 m/s^2 x 2^2) = 50 m - 19.62 m = 30.38 m
Horizontal position = initial horizontal position + (horizontal velocity x time)
Horizontal position = 0 + (30 m/s x 2 s) = 60 m
After 4 seconds:
Vertical velocity = 25 m/s - (9.81 m/s^2 x 4 s) = 25 m/s - 39.24 m/s = -14.24 m/s
(Note: negative sign indicates downward direction)
Vertical position = 0 + (25 m/s x 4 s) - (0.5 x 9.81 m/s^2 x 4^2) = 100 m - 78.48 m = 21.52 m
Horizontal position = 0 + (30 m/s x 4 s) = 120 m
b) To find the time to reach the highest point, we can use the formula for vertical velocity:
Vertical velocity = initial vertical velocity - (acceleration due to gravity x time)
0 = 25 m/s - 9.81 m/s^2 x time
time = 25 m/s / 9.81 m/s^2
time = 2.55 s
c) The height of the highest point can be found using the formula for vertical position:
Vertical position = initial vertical position + (initial vertical velocity x time) - (0.5 x acceleration due to gravity x time^2)
Vertical position = 0 + (25 m/s x 2.55 s) - (0.5 x 9.81 m/s^2 x 2.55 s)^2
Vertical position = 63.75 m
d) The time of flight can be found by doubling the time it took to reach the highest point:
Time of flight = 2.55 s x 2 = 5.1 s
e) The range can be found by multiplying the horizontal velocity by the time of flight:
Range = 30 m/s x 5.1 s = 153 m
Therefore,
a) After 2 seconds, the ball's position is at 60 m horizontally and 30.38 m vertically. After 4 seconds, the ball's position is at 120 m horizontally and 21.52 m vertically.
b) It takes 2.55 seconds to reach the highest point.
c) The highest point is at a height of 63.75 meters.
d) The time of flight is 5.1 seconds.
e) The range of the ball is 153 meters.
A ball is projected with an initial upward velocity component of 25 m/s and a horizontal velocity component of 30 m/s.
a) What are the position and velocity of the ball after 2 s and 4 s?
b) How long a time is required to reach the highest point?
c) How high is this point?
d) What is the time of flight?
e) What is the range?
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