Question

In an arcade game, a ball is launched from the corner of a smooth inclined plane. The inclined plane makes a \(30.0^{\circ}\) angle with the horizontal and has a width of \(w=50.0 \mathrm{~cm}\) The spring-loaded launcher makes an angle of \(45.0^{\circ}\) 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?

Short answer

The given inclined plane has dimensions of \(50.0 \mathrm{~cm}\) width and an inclination angle of \(30^\circ\). A ball is launched from one corner with a launch angle of \(45^\circ\) with the lower edge. Calculate the initial velocity needed for the ball to land in the hole at the opposite corner of the inclined plane.

Step-by-step solution

Determine the horizontal and vertical distances

Given the dimensions of the inclined plane, we need to find the horizontal and vertical distances to the hole from the launching point. The width of the inclined plane is \(50.0 \mathrm{~cm}\), so the horizontal distance \(x\) and the vertical distance \(y\) to the hole can be found using the trigonometric functions for triangles: \(x = w \sin(\theta) = 50 \sin(30^\circ)\) \(y = w \cos(\theta) = 50 \cos(30^\circ)\) where \(\theta = 30^\circ\). Calculate \(x\) and \(y\).

Find the initial horizontal and vertical velocities

Now we need to find the initial horizontal \(v_{0x}\) and vertical velocities \(v_{0y}\) of the ball. As the launcher makes a \(45^\circ\) angle with the lower edge of the inclined plane, we can calculate the initial velocities using: \(v_{0x} = v_0 \cos(45^\circ - \theta)\) \(v_{0y} = v_0 \sin(45^\circ - \theta)\) where \(v_0\) is the initial velocity of the ball, which we are trying to find, and \(\theta = 30^\circ\).

Find the time taken by ball to reach the destination

To find the time taken for the ball to reach the hole, we can consider the horizontal motion. As the horizontal acceleration \(a_x = 0\), the ball will cover the horizontal distance \(x\) with uniform velocity \(v_{0x}\). We can write this as: \(t = \frac{x}{v_{0x}}\).

Set up the equation for the vertical motion and solve for \(v_0\)

We will now consider the vertical motion of the ball. The equation for the vertical motion can be written as: \(y = v_{0y}t - \frac{1}{2} g t^2\) where \(g\) is the acceleration due to gravity. Substituting the expressions for \(t\) and \(v_{0y}\) from the previous step, and using the fact that \(v_{0x} = v_0 \cos(\phi)\), where \(\phi = 45^\circ - \theta\), we get: \(y = v_0 \sin(\phi) \frac{x}{v_0 \cos(\phi)} - \frac{1}{2} g \left(\frac{x}{v_0 \cos(\phi)}\right)^2\). Simplify the equation and solve for \(v_0\). This will give the initial velocity required to launch the ball into the hole at the opposite corner of the inclined plane.

Key concepts

Inclined Plane

An inclined plane is basically a flat surface that is tilted at an angle, which allows objects to be moved up or down with less effort than lifting them directly. In our problem, the inclined plane forms a 30-degree angle with the horizontal. This angle is crucial because it affects how gravity acts on the ball in both directions - vertical and horizontal. When dealing with inclined planes, the force of gravity wants to pull objects straight down, but it also has to drag them along the plane's surface. This creates two components of motion:

• Perpendicular Component: Acts perpendicular to the surface of the plane. This is countered by the plane itself, so it doesn’t affect motion along the plane.
• Parallel Component: Acts along the surface of the plane, pulling the object "down" the slope.

Understanding how an inclined plane works helps in predicting how objects will move along it, which is exactly what we need to calculate the initial velocity of our ball. Knowing the angle helps us decompose the motion into horizontal and vertical parts, both crucial for finding where the ball will land.

Trigonometric Functions

Trigonometric functions are mathematical tools that can help us solve problems involving angles and distances, especially in scenarios involving inclined planes. In our exercise, we use them to calculate the necessary components of motion.The key trigonometric functions we use here are sine (\[\sin\]) and cosine (\[\cos\]). Here's how they help:

• Sine Function: Used to find the horizontal component. It calculates how much of the plane's width contributes to the horizontal distance. In our solution, this is expressed as \[x = w \sin(30^\circ)\], where \[w\] is the width.
• Cosine Function: Used to find the vertical component. It measures the vertical distance contributed by the height of the plane. For our inclined plane, it's represented as \[y = w \cos(30^\circ)\].

Additionally, we use these trigonometric relationships to adjust the motion of the ball from the launcher, which is angled away from the plane. Recognizing the angles and using trigonometry helps us convert between different motion and force components, making this a key tool in solving for the ball's movement.

Initial Velocity Calculation

The calculation of initial velocity is a combination of understanding and applying multiple elements, including the inclined plane and trigonometry. It's crucial as it determines how the ball moves from start to finish, ultimately determining whether it lands on target.Here's the approach:

• Separate Velocity Components: Using the launcher angle, we determine the horizontal (\[v_{0x}\]) and vertical (\[v_{0y}\]) velocity components with respect to the inclined plane by considering the angle of launch.
  For the horizontal component: \[v_{0x} = v_0 \cos(45^\circ - 30^\circ)\]
  For the vertical component: \[v_{0y} = v_0 \sin(45^\circ - 30^\circ)\].
• Calculate Time of Flight: Time (\[t\]) affects how long the ball stays in motion. It can be found using the relation \[t = \frac{x}{v_{0x}}\] since acceleration in the horizontal direction is zero.
• Apply Vertical Motion Formula: The key equation is \[y = v_{0y}t - \frac{1}{2} g t^2\], where \[g\] is gravitational acceleration. By solving this, we find the necessary initial velocity \[v_0\] so that when all the elements come together, the ball will land perfectly inside the hole on the opposite side of the inclined plane.

Finding this initial velocity includes piecing together how energy and motion interact between launch, flight, and landing. It's a comprehensive look at motion dynamics and physics principles in action.