A boat travelling at 30km/h relative to water is headed away from the bank of a river and downstream. The river is 1/2 km wide and flows at 6km/h. The boat arrives at the opposite bank in 1.25min.
Calculate the component of the boat's velocity directed across the river.
Calculate the total downstream component of the boat's motion.
To solve this problem, we need to break down the boat's velocity into its components.
Let's denote the velocity of the boat with respect to the ground as V_b (in km/h) and the velocity of the river as V_r (in km/h).
We are given:
- Velocity of the boat with respect to water (V_bw) = 30 km/h
- Velocity of the river (V_r) = 6 km/h
- Width of the river (W) = 0.5 km
- Time taken to cross the river (t) = 1.25 min = 1.25/60 hours = 0.02083 hours
Now, let's calculate the component of the boat's velocity (V_bx) directed across the river.
V_bx = V_bw - V_r
V_bx = 30 km/h - 6 km/h
V_bx = 24 km/h
Therefore, the component of the boat's velocity directed across the river is 24 km/h.
Next, let's calculate the total downstream component of the boat's motion.
Distance traveled downstream (D_d) = V_bw * t
D_d = 30 km/h * 0.02083 hours
D_d = 0.625 km
Therefore, the total downstream component of the boat's motion is 0.625 km.
To solve this problem, we can use the concept of vector addition to break down the boat's velocity into its horizontal and vertical components.
1. First, let's calculate the component of the boat's velocity directed across the river.
- The boat's velocity relative to the water is given as 30 km/h.
- The river flows at 6 km/h.
- We need to find the component of the boat's velocity perpendicular to the river's flow.
To do this, we can use the Pythagorean theorem. Let v be the boat's velocity across the river, and u be the river's velocity.
The component of the boat's velocity perpendicular to the river's flow can be calculated as:
v^2 = (30 km/h)^2 - (6 km/h)^2
v^2 = (900 km^2/h^2) - (36 km^2/h^2)
v^2 = 864 km^2/h^2
v = sqrt(864) km/h
v ≈ 29.38 km/h
Therefore, the component of the boat's velocity directed across the river is approximately 29.38 km/h.
2. Now, let's calculate the total downstream component of the boat's motion.
- The boat's velocity relative to the water is given as 30 km/h.
- The river flows at 6 km/h.
- We need to find the component of the boat's velocity parallel to the river's flow.
The boat's velocity parallel to the river's flow is equal to the river's velocity, which is 6 km/h.
Therefore, the total downstream component of the boat's motion is 6 km/h.