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In the realm of physics, understanding vector components is vital, especially when dealing with relative motion. Vectors allow us to break down movements into smaller, understandable parts. For this problem, hailstones fall vertically, characterized by a downward vector. When these hailstones hit a sloped windscreen, their velocity can be dissected into two components: horizontal and vertical.
1. **Horizontal Component:** This is the part of the vector acting parallel to the slope of the windscreen. It can be calculated using the sine function, as in the expression: \(v \sin \theta\).
2. **Vertical Component:** Operating perpendicular to the slope, this is determined using the cosine function, represented by \(v \cos \theta\).

By analyzing these components, we can determine how the car's movement interacts with the falling hailstones. Specifically, modifying the car's velocity can align the horizontal component of its motion with the opposite horizontal component created by the slope. This creates the appearance of a vertical bounce.

The velocity ratio \(\frac{v_1}{v_2}\) is a crucial part of understanding relative motion between two objects, here, two cars. This concept involves comparing the speeds of the cars, without delving into their absolute velocities, and makes it easier to understand the problem at hand.

When we express this ratio using the angles of the windscreens and the sine functions, we establish an equation.

• For car 1: Velocity is expressed in terms of the \(30^\circ\) windscreen angle.
• For car 2: Velocity is expressed in terms of the \(15^\circ\) windscreen angle.

This relative comparison through angles allows us to solve for the required speed relationship for the vertical bouncing appearance. Hence, \[ v_1 \sin 30^\circ = v_2 \sin 15^\circ \] is established to equate the horizontal components, leading to a calculated ratio that provides insight into the necessary speeds for both cars.

Trigonometric functions underpin the calculations in this problem scenario, measuring angles to resolve vector components. These functions—particularly sine and cosine—are used to dissect the movement of the hailstones and car's motion in relation to slope angles.

For a windscreen with a given slope angle \(\theta\):

• The **sine function** (\(\sin\)) gives the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle. Here, it helps break down the horizontal component of velocity.
• The **cosine function** (\(\cos\)) provides the ratio of the adjacent side to the hypotenuse, assisting in understanding the vertical component.

These functions allow for precise analysis of the problem: equating the horizontal components of the velocities and thereby enabling the hailstones to appear to bounce vertically. By using specific trigonometric identities and solving these equations, one can simplify complex vector relationships into manageable terms, leading to the solution of the problem.