A stream is 30. meters wide and its current flows southward at 1.5 meters per second. A toy boat is launched with a velocity of 2.0 meters per second eastward from the west bank of the stream
a. what is the magnitude of the boat's resultant velocity as it crosses the stream?
b. how much time is required for the boat to reach the opposite bank of the stream?
To solve this problem, we can use vector addition to find the resultant velocity and then use the formula for time to calculate the time required for the boat to reach the opposite bank of the stream.
Step 1: Determine the horizontal and vertical components of the boat's velocity.
The velocity of the boat can be separated into horizontal (eastward) and vertical (southward) components.
Given:
- Boat's velocity in the eastward direction (horizontal component): 2.0 m/s
- Stream's current velocity in the southward direction (vertical component): 1.5 m/s
Step 2: Calculate the resultant velocity of the boat.
By applying vector addition, we can find the magnitude and direction of the resultant velocity.
The horizontal and vertical components can be combined using the Pythagorean theorem:
Resultant velocity = √((horizontal component)^2 + (vertical component)^2)
Horizontal component = 2.0 m/s (given)
Vertical component = -1.5 m/s (negative sign because it flows southward)
Resultant velocity = √((2.0 m/s)^2 + (-1.5 m/s)^2)
Calculating this further:
Resultant velocity = √(4.0 m^2/s^2 + 2.25 m^2/s^2)
Resultant velocity = √6.25 m^2/s^2
Resultant velocity = 2.5 m/s
Therefore, the magnitude of the boat's resultant velocity as it crosses the stream is 2.5 m/s.
Step 3: Calculate the time required for the boat to reach the opposite bank of the stream.
To calculate the time, we can use the formula:
Time = Distance / Speed
Given:
- Width of the stream: 30 meters
- Resultant velocity of the boat: 2.5 m/s
Since the boat needs to travel from the west bank to the opposite (east) bank of the stream, the horizontal distance to be covered will be 30 meters.
Time = Distance / Speed
Time = 30 meters / 2.5 m/s
Time = 12 seconds
Therefore, it will take the boat 12 seconds to reach the opposite bank of the stream.
To solve this problem, we'll use vector addition to find the resultant velocity of the boat. We'll break down the velocities into their components and then add them together.
a. To find the magnitude of the boat's resultant velocity as it crosses the stream, we need to find the vector sum of the boat's velocity in the eastward direction and the stream's velocity in the southward direction. Let's label the eastward direction as positive x-axis and the southward direction as positive y-axis.
Given:
Boat's velocity (v_boat) = 2.0 m/s (eastward)
Stream's velocity (v_stream) = 1.5 m/s (southward)
To find the magnitude of the boat's resultant velocity (v_resultant), we'll use the Pythagorean theorem:
v_resultant = √(v_x^2 + v_y^2)
v_x = velocity in the x-axis (eastward direction)
v_y = velocity in the y-axis (southward direction)
v_x = 2.0 m/s (eastward) [Given]
v_y = -1.5 m/s (southward) [Negative because it is in the opposite direction of the positive y-axis]
Calculating:
v_resultant = √(2.0^2 + (-1.5)^2)
v_resultant = √(4.0 + 2.25)
v_resultant = √6.25
v_resultant = 2.5 m/s
Therefore, the magnitude of the boat's resultant velocity as it crosses the stream is 2.5 m/s.
b. To find the time required for the boat to reach the opposite bank of the stream, we need to determine the distance the boat needs to travel. We can use the following formula to find the time (t):
distance = velocity × time
The distance the boat needs to travel is the width of the stream (30 meters). Since the boat is crossing the stream at an angle, we can think of it as a right triangle. The time it takes to cross the stream will be the hypotenuse of this triangle.
Using the Pythagorean theorem again, we have:
distance^2 = (velocity_x × time)^2 + (velocity_y × time)^2
Substituting the given values:
(30)^2 = (2.0 × time)^2 + (-1.5 × time)^2
900 = 4.0^2 × time^2 + (-1.5^2) × time^2
900 = 16 × time^2 + 2.25 × time^2
900 = 18.25 × time^2
time^2 = 900 / 18.25
time^2 = 49.32
time = √(49.32)
time = 7.02 seconds (approximately)
Therefore, it would take approximately 7.02 seconds for the boat to reach the opposite bank of the stream.
what is sqrt(1.5^2+2.0^2) ?
timeacross=30/speed across = 30/2.0 sec