A passenger on a stopped bus notices that rain is falling vertically just outside the window. When the bus moves with constant velocity, the passenger observes that the falling raindrops are now making an angle of 18 with respect to the vertical.
What is the ratio of the speed of the raindrops to the speed of the bus?
Find the speed of the raindrops, given that the bus is moving with a speed of 16 .
18deg=arctan(raindropspeed/16)
To solve this problem, we can use the concept of relative velocity. The angle at which the raindrops are observed to fall is due to the combination of the vertical velocity of the raindrops and the horizontal velocity of the bus.
Let's break down the problem step by step:
Step 1: Identify the given information:
- The rain is falling vertically when the bus is stopped.
- The raindrops make an angle of 18 degrees with respect to the vertical when the bus moves with constant velocity.
- The speed of the bus is 16 units (mph, km/h, m/s, etc.).
Step 2: Define the variables:
- Let Vr be the speed of the raindrops (what we need to find).
- Let Vb be the speed of the bus (given as 16).
Step 3: Analyze the problem:
When the bus is moving at a constant velocity, the raindrops will appear to be falling at an angle due to the horizontal velocity of the bus. The vertical component of the raindrops' velocity remains the same.
Step 4: Use trigonometry to relate the vertical and horizontal components:
The tangent of the angle formed by the raindrops with respect to the vertical can be used to relate the vertical and horizontal components of their velocities.
tan(18) = (Vr)/(Vb)
Step 5: Solve for Vr (the speed of the raindrops):
Using trigonometric identities, we can rewrite the equation as:
Vr = Vb * tan(18)
Step 6: Plug in the given values and calculate:
Vr = 16 * tan(18)
Using a scientific calculator, evaluate the tangent of 18 degrees and multiply it by 16 to find the speed of the raindrops.
Calculating the above expression, we find that Vr is approximately 5.72.
Therefore, the ratio of the speed of the raindrops to the speed of the bus is 5.72:16, or simplifying it further, approximately 0.3575.
Step 1: Understand the problem.
The problem describes a situation where a passenger on a moving bus notices the angle at which rain is falling outside the window. The passenger wants to determine the ratio of the speed of the raindrops to the speed of the bus, as well as the speed of the raindrops given that the bus is moving at a speed of 16 units.
Step 2: Identify the given information.
- The angle that the raindrops make with the vertical when the bus is moving is 18 degrees.
- The speed of the bus is 16 units.
Step 3: Determine the ratio of the speed of the raindrops to the speed of the bus.
The key concept to understand is that the horizontal component of the raindrop's velocity is the same as the bus's velocity, while the vertical component is determined by the angle the raindrops make with the vertical. By using trigonometry, we can find this ratio.
Let v_rain be the speed of the raindrops and v_bus be the speed of the bus.
The vertical component of the raindrop's velocity is given by v_rain * sin(18).
Since the raindrops are falling vertically, the vertical component of the bus's velocity is 0.
The horizontal component of the raindrop's velocity is given by v_rain * cos(18).
The horizontal component of the bus's velocity is v_bus.
Since the vertical component of the raindrop's velocity is 0 when the bus is moving, we have the equation:
0 = v_rain * sin(18) - v_bus * sin(0)
Since sin(0) is 0, the equation becomes:
0 = v_rain * sin(18)
This implies that v_rain = 0 or sin(18) = 0.
Since v_rain cannot be equal to 0, we can conclude that sin(18) = 0, which means that v_rain = v_bus * sin(0) / sin(18).
Since sin(0) is 0, this simplifies to:
v_rain = v_bus * sin(0) / sin(18) = v_bus / sin(18)
Hence, the ratio of the speed of the raindrops to the speed of the bus is:
v_rain / v_bus = v_bus / sin(18) / v_bus = 1 / sin(18).
Step 4: Calculate the ratio of the speed of the raindrops to the speed of the bus.
Using a scientific calculator, we can find that sin(18 degrees) is approximately 0.309.
Hence, the ratio of the speed of the raindrops to the speed of the bus is:
v_rain / v_bus = 1 / sin(18) ≈ 1 / 0.309 ≈ 3.24.
Therefore, the ratio of the speed of the raindrops to the speed of the bus is approximately 3.24.
Step 5: Calculate the speed of the raindrops given that the bus is moving at a speed of 16 units.
To find the speed of the raindrops, we can multiply the ratio we found in step 4 by the speed of the bus.
v_rain = ratio * v_bus = 3.24 * 16 = 51.84.
Therefore, the speed of the raindrops is approximately 51.84 units.