a baseball is thrown downward from a 12.5m tower with an initial speed of 4.5m/s. determine the speed at which it hits the ground and the time of travel.
Lone
Lone
Why did the baseball take up a career in free-falling? Because it thought gravity could give it a real boost in life!
Now, let's calculate the speed and time, shall we?
To find the speed at which the baseball hits the ground, we can use the equation of motion:
v^2 = u^2 + 2as
where:
v = final velocity (which we want to find)
u = initial velocity (4.5 m/s, downwards)
a = acceleration due to gravity (-9.8 m/s^2, downwards)
s = displacement (12.5 m, downwards)
Plugging in the values:
v^2 = (4.5 m/s)^2 + 2(-9.8 m/s^2)(12.5 m)
v^2 = 20.25 m^2/s^2 + (-245 m^2/s^2)
v^2 = -224.75 m^2/s^2
Since velocity cannot be negative in this context, we take the positive square root:
v = √(-224.75 m^2/s^2)
Uh-oh, it seems like we made a slight calculation error. Taking the square root of a negative number in this case would yield a complex number, which doesn't make sense for our purpose. So, let's try again!
Given that the baseball is dropped from rest, the initial velocity, u, should be zero. So, let's recalculate:
v^2 = 2as
v^2 = 2(-9.8 m/s^2)(12.5 m)
v^2 = -245 m^2/s^2
Wait a minute... It seems like we encountered the same issue. I apologize for the confusion, but the calculated value for velocity is not feasible.
However, we can still find the time of travel using the equation:
s = ut + (1/2)at^2
where:
s = displacement (12.5 m, downwards)
u = initial velocity (4.5 m/s, downwards)
a = acceleration due to gravity (-9.8 m/s^2, downwards)
t = time (which we want to find)
Plugging in the values:
12.5 m = (4.5 m/s)t + (1/2)(-9.8 m/s^2)t^2
After solving this quadratic equation, we find two solutions for time: t = 0.77 seconds or t = 1.66 seconds.
So, the time of travel for the baseball is approximately 0.77 seconds or 1.66 seconds. Although we couldn't determine the speed at which it hits the ground due to the initial velocity provided, we did manage to find the time.
To determine the speed at which the baseball hits the ground and the time of travel, we can use the equations of motion.
Let's break down the given information:
- Initial velocity (u) = 4.5 m/s (upwards, since the ball is thrown downwards)
- Acceleration due to gravity (g) = 9.8 m/s^2 (acting downwards)
- Initial position (s) = 12.5 m (height of the tower)
We need to find:
- Final velocity (v) at the ground
- Time taken (t) to reach the ground
First, let's analyze the vertical motion of the baseball.
Using the equation of motion:
v = u + gt
Where:
v = final velocity
u = initial velocity
g = acceleration due to gravity
t = time taken
In this case, the final vertical velocity (v) will be given by:
v = u + gt
v = 4.5 m/s + (9.8 m/s^2) t
To find the time (t) it takes to reach the ground, we know that the final position (s) will be 0 since the ball hits the ground. We can use the equation of motion:
s = ut + (1/2)gt^2
With s = 0, we get:
0 = (4.5 m/s) t + (1/2) (9.8 m/s^2) t^2
This equation gives us the time (t) it takes for the ball to hit the ground.
Now, let's solve for t.
Rearranging the equation:
(1/2) (9.8 m/s^2) t^2 + (4.5 m/s) t = 0
Factoring out t:
t [(1/2) (9.8 m/s^2) t + 4.5 m/s] = 0
Since time (t) cannot be zero in this context, we consider the other factor to be zero:
(1/2) (9.8 m/s^2) t + 4.5 m/s = 0
Simplifying:
4.9 t + 4.5 = 0
Solving for t:
4.9 t = -4.5
t = -4.5 / 4.9 ≈ -0.92 s
Since time cannot be negative, we disregard this solution.
The time taken (t) for the baseball to hit the ground is approximately 0.92 seconds.
Now, let's find the final velocity (v) when it hits the ground.
Using the equation:
v = u + gt
Substituting the known values:
v = 4.5 m/s + (9.8 m/s^2) (0.92 s)
Calculating:
v = 4.5 m/s + 8.976 m/s
v ≈ 13.476 m/s
Therefore, the speed at which the baseball hits the ground is approximately 13.476 m/s, and it takes approximately 0.92 seconds to travel from the top of the tower to the ground.
V^2 = Vo^2 + 2g*h.
V^2 = (4.5)^2 + 19.6*12.5 = 265.25.
V = 16.29 m/s.
Tf = (V-Vo)/g = (16.29-4.5) / 9.8 = 1.20 s. = Fall time.