To solve the problem of finding the ratio of the velocities \( v_1 \) and \( v_2 \) of the two cars, we need to analyze the situation involving the hailstones and the cars' windscreen slopes. The key here is to understand the dynamics of elastic collisions and the relative motion of the hailstones with respect to the cars.
Understanding the Scenario
We have two cars moving horizontally with velocities \( v_1 \) and \( v_2 \). Hailstones are falling vertically and, upon hitting the windscreens of the cars, they rebound vertically. This indicates that the horizontal component of the hailstones' velocity must be equal to the horizontal velocity of the cars at the moment of collision.
Analyzing the Collision
In an elastic collision, both momentum and kinetic energy are conserved. However, since the hailstones are falling vertically, we can simplify our analysis by focusing on the horizontal motion. When the hailstones strike the windscreens, they must have a horizontal component of velocity that matches the horizontal velocity of the cars for them to rebound vertically.
Setting Up the Equations
Let’s denote the vertical velocity of the hailstones as \( v_h \) (which is downward). When the hailstones hit the windscreen of the first car, they have a horizontal velocity component of \( v_1 \) (the speed of the first car). For the second car, the same logic applies, and the horizontal component of the hailstones' velocity must equal \( v_2 \).
Since the hailstones rebound vertically, we can conclude that:
- For the first car: The horizontal component of the hailstone's velocity must equal \( v_1 \).
- For the second car: The horizontal component of the hailstone's velocity must equal \( v_2 \).
Finding the Ratio
Since both cars are moving in the same direction and the hailstones rebound vertically, we can establish a relationship between the two velocities. The key insight is that the horizontal velocity of the hailstones must be the same for both cars at the moment of collision. Therefore, we can set up the following relationship:
From the perspective of the hailstones:
- The velocity of the hailstones relative to the first car is \( v_h - v_1 \).
- The velocity of the hailstones relative to the second car is \( v_h - v_2 \).
Since the hailstones rebound vertically, we can equate the magnitudes of these relative velocities:
|v_h - v_1| = |v_h - v_2|.
Assuming \( v_h \) is greater than both \( v_1 \) and \( v_2 \), we can drop the absolute values and rewrite the equation as:
v_h - v_1 = v_h - v_2.
Rearranging gives us:
v_1 = v_2.
Conclusion
Thus, the ratio of the velocities of the two cars is:
v1/v2 = 1
This means that both cars are traveling at the same speed for the hailstones to rebound vertically after colliding with their windscreens. This elegant result highlights the interplay between motion and collision dynamics in a straightforward manner.
