A kayaker paddles due north at 2 m/s. The river the kayaker is paddling in flows towards the east at 1 m/s. After 5 seconds of paddling, how far is the kayaker from where she started in m?
v=sqrt(v₁²+v₂²) = sqrt(4+1) =2.236 m/s
s=vt=2.236•5=11.18 m
To solve this problem, we can use the concept of vector addition.
First, let's analyze the motion of the kayaker. The kayaker is paddling due north at 2 m/s. This motion can be represented by a vector of magnitude 2 m/s pointing directly upwards (northward). Let's call this vector A.
Next, let's consider the motion of the river. The river flows towards the east at 1 m/s. This motion can be represented by a vector of magnitude 1 m/s pointing towards the right (eastward). Let's call this vector B.
Now, we can calculate the resultant of these two vectors, which will give us the net displacement of the kayaker. To do this, we'll add the vectors A and B.
Since the kayaker paddles for 5 seconds, we need to multiply the magnitude of vector A by the time interval. Therefore, the magnitude of vector A becomes 2 m/s * 5 seconds = 10 meters.
To add the vectors, we'll use vector addition. Since the kayaker is paddling due north and the river is flowing towards the east, we need to add vector A and vector B tip-to-tail.
Now, let's break down vector A into its x and y components. Since vector A is pointing directly upwards (northward), the x component of vector A is 0, and the y component is equal to the magnitude of A, which is 10 meters.
For vector B, the x component is equal to the magnitude of B, which is 1 meter, multiplied by the time interval, which is 5 seconds. Therefore, the x component of vector B is 1 m/s * 5 s = 5 meters. The y component of vector B is 0 since it does not contribute to the kayaker's displacement in the vertical direction.
To find the resultant vector, we add the x and y components of vectors A and B.
The x component: 0 + 5 = 5 meters
The y component: 10 + 0 = 10 meters
Using the Pythagorean theorem, we can calculate the magnitude of the resultant vector:
Magnitude = sqrt(5^2 + 10^2) = sqrt(25 + 100) = sqrt(125) = 11.18 meters
Therefore, after 5 seconds of paddling, the kayaker is approximately 11.18 meters away from the starting point.