A car is traveling horizontally at 15.0 mph; raindro
ps have a speed of 20.0
mph with respect to the
car. A passenger in the car notices
the rain
is slanted
at 30
°
from vertical toward the back of the
car. Find the velocity of the rain with respect to the ground
To find the velocity of the rain with respect to the ground, we need to consider the velocity vectors of the car and the raindro ps separately and then add them together.
1. Convert the car's speed to meters per second (m/s):
15.0 mph = (15.0 * 0.44704) m/s = 6.7056 m/s
2. Decompose the raindro p's velocity into vertical and horizontal components:
The rain is slanted at 30° toward the back of the car, so the vertical component of the raindro p's velocity is 20.0 mph * sin(30°) = 20.0 * 0.5 = 10.0 mph
The horizontal component of the raindro p's velocity remains the same: 20.0 mph * cos(30°) = 20.0 * 0.866 = 17.32 mph
3. Convert the raindro p's components to meters per second (m/s):
Vertical component: 10.0 mph = (10.0 * 0.44704) m/s = 4.4704 m/s
Horizontal component: 17.32 mph = (17.32 * 0.44704) m/s = 7.740608 m/s
4. Add the horizontal components of the car and raindro p velocities:
6.7056 m/s + 7.740608 m/s = 14.446208 m/s
5. The velocity of the raindro p with respect to the ground is 14.446208 m/s horizontally and 4.4704 m/s vertically.
To find the velocity of the rain with respect to the ground, we need to break down the velocities into their horizontal and vertical components.
Let's first convert the speeds from mph to m/s to maintain consistent units. Since 1 mph is equal to 0.447 m/s, we get:
Car velocity (v_car) = 15.0 mph = 15.0 * 0.447 m/s = 6.68 m/s
Rain velocity with respect to the car (v_rain_car) = 20.0 mph = 20.0 * 0.447 m/s = 8.94 m/s
Now, let's analyze the components of the rain velocity relative to the ground.
The vertical component of the rain velocity (v_rain_y) is determined by the slant angle of 30°. Since the angle is measured from the vertical and is slanted towards the back of the car, the vertical component would be negative. Therefore,
v_rain_y = -v_rain_car * sin(30°)
= -8.94 m/s * sin(30°)
≈ -4.47 m/s
The horizontal component of the rain velocity (v_rain_x) remains unaffected by the slant angle and is equal to the car's velocity,
v_rain_x = v_car
= 6.68 m/s
Finally, we can compute the velocity of the rain with respect to the ground by combining the horizontal and vertical components. Since velocity is a vector, we can use the Pythagorean theorem to find its magnitude, and the inverse tangent function to find its direction.
Magnitude of rain velocity with respect to ground (v_rain_ground) = sqrt(v_rain_x^2 + v_rain_y^2)
= sqrt((6.68 m/s)^2 + (-4.47 m/s)^2)
≈ 7.98 m/s
Direction of rain velocity with respect to ground = atan(v_rain_y / v_rain_x)
= atan((-4.47 m/s) / (6.68 m/s))
≈ -34.33°
Therefore, the velocity of the rain with respect to the ground is approximately 7.98 m/s, making an angle of about -34.33° from the horizontal. Note that the negative sign indicates the direction of the rain is opposite to the car's motion.