A ball is thrown upwards. Neglecting air resistance what initial speed does the ball need to remain in the air for a total time of 10 seconds? PLEASE show work :)
Tr = 10/2 = 5 s. = Rise time.
V = Vo + g*Tr = 0.
Vo + (-9.8)5 = 0,
Vo =
To find the initial speed of the ball, we need to consider the kinematic equation for vertical motion:
h = v₀t + (1/2)gt²
Where:
h = vertical displacement (0 since the ball starts and ends at the same height)
v₀ = initial velocity (which is what we need to find)
t = total time in the air (10 seconds)
g = acceleration due to gravity (approximately -9.8 m/s²)
Since the ball is thrown upwards, the initial velocity v₀ should be positive. So, the equation becomes:
0 = v₀ * 10 + (1/2)(-9.8)(10)²
To solve for v₀, we can rearrange the equation as:
v₀ = -5 * (-9.8)(10)
v₀ = 490 m/s
Therefore, the ball needs an initial speed of 490 m/s to remain in the air for a total time of 10 seconds, neglecting air resistance.
To determine the initial speed required for the ball to remain in the air for a total time of 10 seconds, we can use the kinematic equations of motion. Specifically, we'll use the equation:
d = vit + 0.5at^2
where:
d is the displacement (change in height),
vi is the initial velocity (speed),
t is the time, and
a is the acceleration (in this case, due to gravity).
Since the ball is being thrown upwards, the acceleration due to gravity will be negative (-9.8 m/s^2), as it acts opposite to the direction of motion.
Let's assume that the ball reaches its maximum height at time t1, and then falls back to the initial height at time t2. Therefore, the total flight time is given by:
t_total = t1 + t2
Since the ball reaches its maximum height when its vertical velocity becomes zero, we'll have:
vi - 9.8t1 = 0
Solving for t1:
t1 = vi / 9.8
Similarly, the total flight time is the sum of the time going up and coming down, so:
t_total = 2t1
Let's substitute back the equation for t1:
t_total = 2(vi / 9.8)
Now, we can rearrange this equation to solve for vi:
vi = (9.8 * t_total) / 2
Plugging in the given value of t_total = 10 seconds:
vi = (9.8 * 10) / 2
Calculating this:
vi = 49 m/s
Therefore, the initial speed the ball needs to remain in the air for 10 seconds is 49 m/s.