#4. Rain is falling at a velocity of 10,0 m/s at an angle of 30* towards the west. At what velocity and direction does a marathon runner have to go in order for him to get rain on his head?
Since the rain is falling with some nonzero downward speed, there is no way the runner can avoid getting rain on his head.
If he runs at just the right speed, the rain will appear to be falling straight down on him.
At any other speed, rain will hit him either in the face or the back of the head, at some angle.
To determine the velocity and direction at which the marathon runner needs to move in order to get rain on their head, we need to consider the relative motion between the runner and the rain.
First, let's break down the given information:
- Rain velocity: 10.0 m/s
- Angle of rain: 30 degrees towards the west
Now, let's analyze the situation step by step:
1. Convert the angle from westward to northward motion. Since west is 90 degrees counterclockwise from north, we can subtract 30 degrees from 90 degrees to get 60 degrees northward.
2. Using the concept of vector addition, we need to determine the component of the rain's velocity directed towards the north. To do this, we can use trigonometry. The northward component can be calculated as follows:
Velocity northward = Rain velocity * sin(angle)
Velocity northward = 10.0 m/s * sin(60 degrees)
Velocity northward = 10.0 m/s * 0.866 (approximately)
Velocity northward = 8.66 m/s (approximately)
3. To counteract the rain's northward velocity, the runner needs to run southward with the same speed but in the opposite direction. Therefore, the velocity at which the runner needs to move is 8.66 m/s southward.
4. Finally, to determine the direction, we need to specify the angle relative to a reference direction. Let's consider the reference direction as north. Since the runner needs to move southward, the angle would be 180 degrees south.
Therefore, the marathon runner needs to move at a velocity of 8.66 m/s towards the south (opposite to the rain's northward velocity) to get rain on their head. The direction can be specified as 180 degrees south.