You are riding in a boat whose speed relative to the water is 4.5 m/s. The boat points at an angle of 25.1¡ã upstream on a river flowing at 12.8 m/s. Find the time it takes for the boat to reach the opposite shore if the river is 26.2 m wide.
d = 26.2/cos25 = 28.91 m. = Distance the
boat must travel to reach shore.
T = d/r = 28.91m./4.5m/s = 6.42 s.
To find the time it takes for the boat to reach the opposite shore, we need to break down the motion of the boat into its components.
Let's consider the horizontal and vertical components separately.
Horizontal Component:
The horizontal component includes the motion of the boat parallel to the river flow. Since the boat is pointing upstream, the component of the boat's velocity in the x-direction will be the sum of its speed relative to the water and the river's velocity.
Given:
Boat's speed relative to water = 4.5 m/s
River's velocity = 12.8 m/s
Using basic trigonometry, we can find the horizontal component of the boat's velocity:
Horizontal Component of Boat's Velocity = Boat's Speed relative to Water * cos(angle)
Horizontal Component of Boat's Velocity = 4.5 m/s * cos(25.1°)
Vertical Component:
The vertical component includes the motion of the boat perpendicular to the river flow. Since the boat is pointing upstream, the vertical component of the boat's velocity will be the difference between its speed relative to the water and the river's velocity.
Using basic trigonometry, we can find the vertical component of the boat's velocity:
Vertical Component of Boat's Velocity = Boat's Speed relative to Water * sin(angle)
Vertical Component of Boat's Velocity = 4.5 m/s * sin(25.1°)
Now that we have the horizontal and vertical components of the boat's velocity, we can calculate the time it takes for the boat to cross the river.
Time = Distance / Horizontal Component of Boat's Velocity
Given:
Distance = 26.2 m (width of the river)
Plugging in the values:
Time = 26.2 m / (4.5 m/s * cos(25.1°))
Solve for time to get the answer.
To find the time it takes for the boat to reach the opposite shore, we can break down the boat's velocity into its horizontal and vertical components.
First, let's find the vertical component of the velocity. Since the boat is pointed upstream, we know that this component will be against the river flow. We can use the sine function to find this component:
Vertical component = Boat speed x sin(angle)
Vertical component = 4.5 m/s x sin(25.1°)
Next, let's find the horizontal component of the velocity. This component will be parallel to the river flow. We can use the cosine function to find this component:
Horizontal component = Boat speed x cos(angle)
Horizontal component = 4.5 m/s x cos(25.1°)
Now, we have the horizontal and vertical components of the boat's velocity. The vertical component will determine how long it takes for the boat to cross the river because it is in the direction perpendicular to the river flow.
Since the width of the river is given as 26.2 m, we can divide the width by the vertical component of the boat's velocity to find the time it takes to cross the river.
Time = River width / Vertical component
Time = 26.2 m / (4.5 m/s x sin(25.1°))
Now, you can calculate the time it takes for the boat to reach the opposite shore using the given values.