A ball is thrown horizontally from the top of a building.

The ball is thrown with a horizontal speed of 8.2 ms^-1. The side of the building is vertical. At point P on the path of the ball, the ball is distance x from the building and is moving at an angle of 60 degree to the horizontal. Air resistance is negligible.

a. For the ball at point P.
i. show that the vertical component of it's velocity is 14.2 ms^-1.
ii.determine the vertical distance through which ball has fallen
iii.determine the horizontal distance x.

do the vertical, z, and horizontal , x , problems separately.

They are connected by location and time.
At start Vx = 8.2 and Vz = 0
There are no horizontal forces so Vx = 8.2 forever, or until a crash.
That means x = 8.2 t forever.

Now at V = 60 degrees down from horizontal Vx = |V| cos 60 still = 8.2
so
8.2 = |V| * 0.5
so then |V| = 16.4
and Vz = -|V| sin 60 = -16.4 * .866 = -14.2 sure enough (note I call z positive up so it is negative)
now the vertical problem:
at top z = 0 and Vz = 0
Vz = -g t = -9.81 t
so
-14.2 = -9.81 t
and
t = 1.45 seconds
and
z = 0 + 0 t - 4.9 t^2
z = -4.9 * (1.45)^2 = - 10.3 meters
so it fell 10.3 meters
x = 8.2 forever remember :)
x = 8.2 * 1.45 meters

For Vertical Velocity:

Tanθ =x/8.2 m/s^2
X=8.2 * √ 3 = 14.2 m/s^2. It is downward, so -14.2 m/s^2

To measure the time, divide velocity by acceleration
a=v/t
t=v/a
t=-14.2 m/s / -9.8 m/s^2
t= 1.45 s

For distance, multiply velocity by time
d= v*t
d= 14.2 m/s * 1.45 s
d=20.59 m

To solve this problem, we can use the principles of projectile motion.

a.i. To find the vertical component of the velocity at point P, we can use trigonometry. The angle of the velocity vector with the horizontal is given as 60 degrees.

Using the given horizontal speed of 8.2 m/s and the angle of 60 degrees, we can find the vertical component of the velocity using the equation:

Vertical component of velocity = 8.2 m/s * sin(60 degrees)

Evaluating this expression:

Vertical component of velocity = 8.2 m/s * 0.866

Vertical component of velocity = 7.0972 m/s (rounded to 4 decimal places)

Therefore, the vertical component of the velocity at point P is 7.0972 m/s.

a.ii. To determine the vertical distance through which the ball has fallen, we can use the equation for vertical displacement.

The formula for vertical displacement (or vertical distance) is given by:

Vertical displacement = (1/2) * g * t^2

where g is the acceleration due to gravity (approximately 9.8 m/s^2) and t is the time of flight.

Since the ball is thrown horizontally, we know that the time of flight is the same for both the horizontal and vertical motion.

The time of flight can be found using the equation:

Time of flight = horizontal distance / horizontal velocity

Since the horizontal distance is x and the horizontal velocity is 8.2 m/s, the time of flight is:

Time of flight = x / 8.2

Using the equation for vertical displacement, and substituting the known values:

Vertical displacement = (1/2) * 9.8 * (x / 8.2)^2

Simplifying this expression:

Vertical displacement = 1.1837 * x^2 (rounded to 4 decimal places)

Therefore, the vertical distance through which the ball has fallen is 1.1837 times x^2.

a.iii. To determine the horizontal distance x, we can use the equation for horizontal displacement.

Since the horizontal motion is at a constant velocity, we can find the horizontal distance using the equation:

Horizontal distance = horizontal velocity * time of flight

Substituting the given values:

Horizontal distance = 8.2 * (x / 8.2)

Simplifying this expression:

Horizontal distance = x

Therefore, the horizontal distance x is equal to x.

To answer these questions, we need to break down the problem into horizontal and vertical components. Let's use the equations of motion to find the answers.

a. i. To find the vertical component of velocity at point P, we can use the trigonometric relationship of the given angle. The vertical component can be found using the equation:

Vertical component of velocity = Horizontal speed * sin(angle)

Given:
Horizontal speed (u) = 8.2 m/s
Angle (θ) = 60 degrees

Using the equation, we have:
Vertical component of velocity = 8.2 m/s * sin(60 degrees)
Vertical component of velocity = 8.2 m/s * √3/2
Vertical component of velocity = 14.2 m/s

Therefore, the vertical component of velocity at point P is 14.2 m/s.

a. ii. To determine the vertical distance through which the ball has fallen, we can use the equation of motion:

Vertical distance (s) = (Vertical component of velocity)^2 / (2 * acceleration due to gravity)

Given:
Vertical component of velocity (v) = 14.2 m/s
Acceleration due to gravity (g) = 9.8 m/s^2

Using the equation, we have:
Vertical distance = (14.2 m/s)^2 / (2 * 9.8 m/s^2)
Vertical distance ≈ 10.274 m

Therefore, the vertical distance through which the ball has fallen is approximately 10.274 meters.

a. iii. To determine the horizontal distance (x), we can use the equation of motion for horizontal motion:

Horizontal distance (x) = Horizontal speed * Time

In this problem, we need to find the time it takes for the ball to reach point P. Since we know the horizontal speed and acceleration due to gravity does not affect horizontal motion, the time taken to reach point P will be the same as if the ball was projected horizontally with no vertical motion.

Time = Distance / Horizontal speed

Given:
Horizontal speed (u) = 8.2 m/s

Since the ball was thrown horizontally from the top of the building, it has the same horizontal displacement (x) as the distance from the building.

Therefore, the horizontal distance (x) is directly equal to the time taken to reach point P.

Horizontal distance (x) = 8.2 m/s * Time

To find the time taken, we need to consider the vertical motion of the ball. The time taken for the vertical motion is the same as the time taken to reach the peak height before falling back down. The formula for time taken for vertical motion is:

Time = Vertical component of velocity / acceleration due to gravity

Using the given values:
Vertical component of velocity (v) = 14.2 m/s
Acceleration due to gravity (g) = 9.8 m/s^2

Time = 14.2 m/s / 9.8 m/s^2
Time ≈ 1.45 seconds

Using the time to reach point P, we can determine the horizontal distance:

Horizontal distance (x) = 8.2 m/s * 1.45 s
Horizontal distance (x) ≈ 11.89 meters

Therefore, the horizontal distance (x) from the building at point P is approximately 11.89 meters.