A cyclist who is travelling directly to the East at 8 kilometres per hour
finds that the wind appears to blow directly from the North. She then continues to
travel East but she doubles her speed to 16 kilometers per hour and the wind then
appears to blow from the North-East. Find the actual velocity of the wind.
Vector makes a Triangle
a^2+b^2=c^2
a=b=8 km/h
8^2+8^2=c^2
solve for C
To find the actual velocity of the wind, we can analyze the situation using vector addition.
Let's denote the actual velocity of the cyclist as v_c and the actual velocity of the wind as v_w. We also need to consider the apparent velocity of the wind, which is the relative velocity of the wind as observed by the cyclist.
In the first case, when the cyclist is traveling east at 8 km/h and the wind appears to blow directly from the north, we can set up the following equations:
v_c + v_w = 0 (Equation 1)
Apparent wind velocity = v_w - v_c = northward velocity
Since the apparent wind velocity is directly northward, we can assume that the northward component of the wind velocity is equal to the magnitude of the cyclist's velocity.
So we can rewrite Equation 1 as follows:
v_c + v_w = 0 (Equation 1)
v_c + | v_w | = 8 (Equation 2)
Now, let's consider the second case, when the cyclist doubles her speed to 16 km/h and the wind appears to blow from the northeast. We can set up the following equations:
v_c + v_w = 0 (Equation 1)
Apparent wind velocity = v_w - v_c = northeastward velocity
Since the apparent wind velocity is northeastward, we can assume that the northward and eastward components of the wind velocity are equal.
So we can rewrite Equation 1 as follows:
v_c + v_w = 0 (Equation 1)
v_c + | v_w | * (√2) = 16 (Equation 3)
Now we have a system of equations (Equations 2 and 3) that we can solve to find the values of v_c and |v_w|, which will give us the actual velocity of the wind.
Let's solve the system of equations:
Subtracting Equation 2 from Equation 3:
| v_w | * (√2) - | v_w | = 16 - 8
(| v_w | * √2) - | v_w | = 8
Simplifying further:
| v_w | (√2 - 1) = 8
Now we can solve for | v_w |:
| v_w | = 8 / (√2 - 1)
Therefore, the magnitude of the wind velocity is | v_w | = 22.61 km/h.
To find the actual velocity of the wind, we need to consider both magnitude and direction. Since the wind velocity was initially from the north and then shifted towards the northeast, we can assume the wind velocity vector forms a right angle with the cyclist's velocity vector.
Hence, the actual velocity of the wind is v_w = 22.61 km/h at a direction of 45 degrees northeastward.