a) Rain drops hitting the side window of a car in motion often leave diagonal streaks if there is no wind. Why?
b) For a certain projectile, the horizontal range is twice in magnitude of the maximum height attained by it. Find its angle of projection.
a) The diagonal streaks left by raindrops hitting the side window of a car in motion occur due to the combination of the raindrop's velocity and the car's motion. When the car is moving, it creates a relative motion between the raindrops and the window. Since there is no wind, the raindrops have no horizontal component of velocity relative to the car. However, they do have a vertical component of velocity due to gravity.
As the raindrops fall, the vertical component of their velocity causes them to fall diagonally towards the ground. At the same time, the horizontal motion of the car makes the raindrops hit the window at an angle. This combination of vertical and horizontal motion results in the diagonal streaks you observe on the window.
b) To find the angle of projection of the projectile, we need to consider the projectile motion equations. Let's assume that the angle of projection is θ and the initial velocity is v.
Given that the horizontal range (R) is twice the magnitude of the maximum height (H), we can write the following equations:
R = 2H ---(1)
The horizontal range can be expressed in terms of the initial velocity (v), angle of projection (θ), and the acceleration due to gravity (g):
R = v^2 * sin(2θ) / g ---(2)
The maximum height can be expressed as:
H = v^2 * sin^2(θ) / (2g) ---(3)
Substituting equations (2) and (3) into equation (1), we get:
v^2 * sin(2θ) / g = 2 * v^2 * sin^2(θ) / (2g)
Simplifying, we have:
sin(2θ) = 2 * sin^2(θ)
Using the double-angle identity for sine (sin(2θ) = 2sin(θ)cos(θ)), we can rewrite the equation as:
2sin(θ)cos(θ) = 2 * sin^2(θ)
Dividing both sides by 2sin(θ), we get:
cos(θ) = sin(θ)
Since cos(θ) = sin(90° - θ), we have:
90° - θ = θ
Solving for θ, we find:
2θ = 90°
θ = 45°
Therefore, the angle of projection is 45 degrees.