To a cyclist riding west at 20 kg/hr, the rain appears to meet him at an angle of 45° with the vertical. When he rides at 12 km/hr, the rain meets him at an angle of 19°48’ with the vertical. What is the actual direction of the rain?
R is speed of rain
r is angle of rain to vertical
horizontal rain we see = Rsinr+bike speed
vertical rain we see = R cos r
case 1
hor = R sin r + 20
ver = R cos r
tan 45 = (R sin r +20) /Rcos r
case 2
tan 19deg48 min = (R sinr +12)/R cos r
To find the actual direction of the rain, we can use the concept of relative velocity.
Let's assume that the cyclist is riding in a calm condition (no wind affecting the motion of the cyclist or raindrops) and that the direction in which the cyclist is riding is west.
When the cyclist rides at 20 km/hr (20 kg/hr) and the rain appears to meet him at an angle of 45° with the vertical, it means that the rain is falling vertically downward with respect to the cyclist's frame of reference. This is because the angle of 45° indicates that the cyclist is experiencing no horizontal effect due to raindrops.
Now, when the cyclist rides at 12 km/hr, and the rain meets him at an angle of 19°48’ with the vertical, it means that the raindrops have a horizontal component of motion relative to the cyclist's frame of reference. This change in angle is due to the horizontal velocity of the cyclist.
To find the actual direction of the rain, we need to calculate the horizontal component of the rain's velocity relative to the ground. We can achieve this by comparing the horizontal velocities of the cyclist and the raindrops.
Let's calculate the horizontal component of the rain's velocity:
The horizontal component of the rain's velocity relative to the cyclist's frame of reference at 12 km/hr is:
Horizontal velocity of raindrops = 12 km/hr * sin(19°48’)
If we convert this value to the same unit as the velocity of the cyclist, which is km/hr:
Horizontal velocity of raindrops = 12 km/hr * sin(19°48’) = 12 km/hr * 0.3354 ≈ 4.0248 km/hr
Now, to find the actual direction of the rain, we need to consider the horizontal velocity of the cyclist:
Horizontal velocity of cyclist = 12 km/hr
Since the raindrops have a horizontal velocity component of 4.0248 km/hr with respect to the cyclist, the actual direction of the rain would be the vector sum of the horizontal velocity of the raindrops and the horizontal velocity of the cyclist.
Calculating the vector sum of these velocities:
Actual direction of rain = arctan(4.0248 km/hr / 12 km/hr)
Using a calculator, we find:
Actual direction of rain ≈ 18.4343°
Therefore, the actual direction of the rain is approximately 18.4343° west of the vertical.