In an arcade game, a ball is launched from the corner of a smooth inclined plane. The inclined plane makes a 33.3° angle with the horizontal and has a width of w = 42.9 cm. The spring-loaded launcher makes an angle of 45.0° with the lower edge of the inclined plane. The goal is to get the ball into a small hole at the opposite corner of the inclined plane. With what initial velocity should you launch the ball to achieve this goal?

To determine the initial velocity needed to launch the ball into the hole, we can use the principles of projectile motion.

Let's break down the problem into steps and use the following variables:
- θ1: The angle of the inclined plane (33.3°)
- θ2: The angle of the spring-loaded launcher (45.0°)
- w: The width of the inclined plane (42.9 cm)
- g: Acceleration due to gravity (9.8 m/s²)
- v0: The initial velocity of the ball

Step 1: Find the horizontal distance traveled by the ball on the inclined plane.
We can use the trigonometric relationship between the angle θ1 and the width w to find the horizontal distance (d) traveled by the ball on the inclined plane.
d = w * sin(θ1)

Step 2: Find the time of flight on the inclined plane.
Since we want the ball to reach the hole at the opposite corner, we can calculate the time it takes for the ball to reach that point using the horizontal distance and the horizontal component of the initial velocity.
The time of flight on the inclined plane (t1) is given by:
t1 = d / (v0 * cos(θ2))

Step 3: Find the vertical distance traveled by the ball on the inclined plane.
Using the horizontal distance d, the angle θ1, and the trigonometric relationship, we can find the vertical distance (h) traveled by the ball on the inclined plane.
h = d * tan(θ1)

Step 4: Find the initial velocity of the ball.
Now, we can use the vertical distance h and the time of flight t1 to calculate the initial velocity of the ball required for it to travel the vertical distance and reach the hole.
Using the equation of motion for vertical displacement (h):
h = (1/2) * g * t1²

Rearranging the equation, we can solve for the initial velocity (v0):
v0 = √((2 * h * g) / t1²)

Finally, substitute the values into the equation and solve for v0.

Note: Make sure to convert all units into consistent units before substituting them into the equations.

I hope this helps you understand the process of solving this problem!

To find the initial velocity needed to launch the ball into the small hole, we can use the concept of projectile motion.

Step 1: Determine the components of the initial velocity.
The initial velocity can be broken down into its horizontal and vertical components.

The horizontal component can be found using the angle between the inclined plane and the horizontal:
horizontal component = velocity * cos(angle)

The vertical component can be found using the angle between the inclined plane and the horizontal:
vertical component = velocity * sin(angle)

Step 2: Calculate the time of flight.
The time it takes for the ball to reach the opposite corner can be found using the vertical component of the initial velocity and the acceleration due to gravity:
time of flight = (2 * vertical component) / g

Step 3: Calculate the horizontal distance.
The horizontal distance the ball needs to travel can be calculated using the width of the inclined plane and the time of flight:
horizontal distance = width

Step 4: Calculate the horizontal component of the initial velocity.
The horizontal component of the initial velocity can be found by rearranging the equation for horizontal distance:
initial velocity * time of flight * cos(angle) = horizontal distance
initial velocity = horizontal distance / (time of flight * cos(angle))

Step 5: Substitute values and solve.
Substitute the values given in the problem into the equations above:

angle = 33.3°
width = 42.9 cm = 0.429 m
g = 9.8 m/s^2

First, calculate the vertical component:
vertical component = velocity * sin(angle)

Then, calculate the time of flight:
time of flight = (2 * vertical component) / g

Next, calculate the horizontal distance:
horizontal distance = width

Finally, calculate the initial velocity:
initial velocity = horizontal distance / (time of flight * cos(angle))

Plug in the values:
vertical component = initial velocity * sin(angle)
time of flight = (2 * vertical component) / g
horizontal distance = width
initial velocity = horizontal distance / (time of flight * cos(angle))

Solving these equations step-by-step will give you the initial velocity needed to launch the ball into the small hole.