A ball shot at an angle of 60° to the ground strikes a building 23m away at a point 16m

high. Find the magnitude and direction of the final velocity of the ball as it strikes the wall.

Angle of elevation 60

initial speed = s
initial vertical speed Vi
= s sin 60 = .866 s
horizontal speed forever = u = s cos 60
= s/2

t = 23/u = 46/s

at t = 46/s, h = 16

16 = 0 + Vi t - 4.9 t^2
16 = .866 s (46/s) - 4.9*46^2/s^2
-23.8 = -10368/s^2
s^2 = 435.6
s = 20.9 m/s
That is INITIAL speed
now t = 46/s = 2.2 seconds
Vi = .866 * 20.9 = 18.1
u = .5 * 20.9 = 10.5
v = 18.1 - 9.8 (2.2) = - 3.46
magnitude of speed = sqrt(u^2+v^2)
= sqrt (10.5^2 + 3.46^2)
= 11.1 m/s
tan angle down from horizontal = 3.46/10.5
so angle down from horizontal = 18.2 degrees downward

To find the magnitude and direction of the final velocity of the ball, we can split the initial velocity of the ball into horizontal and vertical components.

Given:
Initial angle of 60°
Horizontal distance of 23m
Vertical distance of 16m

Step 1: Horizontal Velocity
The horizontal component of the initial velocity (Vx) remains constant throughout the motion. We can find it using the equation:

Vx = V * cosθ

where:
V: magnitude of the initial velocity
θ: angle of projection (in radians)

The given angle is 60°, so we need to convert it to radians:

θ = 60° * (π/180) = π/3 radians

Now we can calculate the horizontal component:

Vx = V * cos(π/3)

Step 2: Vertical Velocity
The vertical component of the initial velocity (Vy) changes due to the effect of gravity. We can find it using the equation:

Vy = V * sinθ

where:
V: magnitude of the initial velocity
θ: angle of projection (in radians)

Using the given angle, we can calculate the vertical component:

Vy = V * sin(π/3)

Step 3: Time of Flight
The time it takes for the ball to reach the building can be determined using the vertical component of the motion. We can use the equation:

h = Vyt + (1/2)gt^2

where:
h: vertical distance (16m)
Vy: vertical component of the initial velocity
t: time of flight
g: acceleration due to gravity (approximately 9.8 m/s²)

Substituting the known values, we get:

16 = V * sin(π/3) * t + (1/2) * 9.8 * t^2

This is a quadratic equation, and by solving it, we can find the value of t.

Step 4: Final Velocity
To find the final velocity of the ball, we need to combine the horizontal and vertical components. Given that the time of flight is now known, we can determine the horizontal distance traveled (23m).

Using the following equation:

Horizontal Distance = Vx * t

We can solve for Vx:

23 = V * cos(π/3) * t

Step 5: Magnitude and Direction of Final Velocity
The magnitude of the final velocity (Vf) is given by:

Vf = √(Vx^2 + Vy^2)

The direction (angle) of the final velocity (θf) is given by:

θf = tan^(-1)(Vy/Vx)

Plugging in the values for Vx and Vy that we calculated earlier, we can find the magnitude and direction of the final velocity of the ball as it strikes the wall.

To find the magnitude and direction of the final velocity of the ball as it strikes the wall, we can use the principles of projectile motion.

Step 1: Split the initial velocity into horizontal and vertical components.
The initial velocity can be split into two components: the horizontal component (Vx) and the vertical component (Vy). Since the ball is shot at an angle of 60° to the ground, we can calculate these components using trigonometry.

Vx = V * cos(θ)
where V is the initial velocity of the ball and θ is the angle of projection.
Vy = V * sin(θ)

Step 2: Determine the time of flight.
The time it takes for the ball to travel horizontally to the building can be calculated using the horizontal component of velocity.
t = distance / Vx
where distance is the horizontal distance to the building.

Step 3: Find the vertical distance traveled by the ball.
Using the vertical component of the velocity, we can calculate the vertical distance traveled by the ball during the time of flight.
y = Vy * t - 0.5 * g * t^2
where g is the acceleration due to gravity (approximately 9.8 m/s^2).

Step 4: Calculate the final velocity.
The final vertical velocity (Vfy) can be calculated by subtracting the initial vertical velocity (Vy) from the product of gravity and the time of flight.
Vfy = Vy - g * t

The final velocity (Vf) can be calculated using the Pythagorean theorem.
Vf = sqrt(Vfx^2 + Vfy^2)
where Vfx is the horizontal component of the final velocity.

Step 5: Calculate the direction of the final velocity.
The angle (θf) of the final velocity with respect to the ground can be calculated using the inverse tangent function.
θf = arctan(Vfy / Vfx)

Plugging in the known values and performing the calculations will give us the magnitude and direction of the final velocity.