A ball is thrown horizontally from the top of
a building 45.2 m high. The ball strikes the
ground at a point 78.9 m from the base of the
building.
The acceleration of gravity is 9.8 m/s
2
.
Find the time the ball is in motion, initial velocity, and x and y components
To solve this problem, we can use the equations of motion to find the time the ball is in motion, the initial velocity, and the x and y components.
First, let's find the time the ball is in motion. Since the ball is thrown horizontally, the initial vertical velocity is 0. We can use the equation for free fall motion:
y = y0 + v0yt + (1/2)gt^2
Where:
y = final vertical position = 0 (since the ball hits the ground)
y0 = initial vertical position = 45.2 m
v0y = initial vertical velocity = 0 m/s
g = acceleration due to gravity = 9.8 m/s^2
t = time
By substituting the values into the equation, we get:
0 = 45.2 + 0*t - (1/2)*9.8*t^2
Simplifying the equation, we have:
0 = 45.2 - 4.9t^2
Rearranging the equation, we get a quadratic equation:
4.9t^2 - 45.2 = 0
To solve for t, we can use the quadratic formula:
t = (-b ± √(b^2 - 4ac))/(2a)
In this case, a = 4.9, b = 0, and c = -45.2. Substituting the values, we get:
t = (-0 ± √(0^2 - 4*4.9*(-45.2)))/(2*4.9)
Simplifying the equation gives us two values for t:
t = 3.38 s or t = -3.38 s
Since time cannot be negative, the ball is in motion for approximately 3.38 seconds.
Now, let's find the initial velocity. Since the horizontal velocity remains constant throughout the motion, we can use the equation:
x = x0 + v0xt
Where:
x = final horizontal position = 78.9 m
x0 = initial horizontal position = 0 (since the ball is thrown from the top of the building)
v0x = initial horizontal velocity = ?
By substituting the values into the equation, we get:
78.9 = 0 + v0x * t
78.9 = v0x * 3.38
Simplifying the equation, we get:
v0x = 78.9 / 3.38
v0x = 23.3 m/s
Therefore, the initial velocity in the x-component is 23.3 m/s.
Lastly, let's find the y-component of the initial velocity. Since the ball is thrown horizontally, the vertical velocity remains constant at 0 m/s throughout the motion.
Therefore, the initial velocity in the y-component is 0 m/s.
To summarize:
- The time the ball is in motion is approximately 3.38 seconds.
- The initial velocity in the x-component is 23.3 m/s.
- The initial velocity in the y-component is 0 m/s.
the height y is
y = 45.2 - 4.9t^2
solve for t when y=0
using that t, since the horizontal velocity v does not change,
vt = 78.9