Mary leaves a dock paddling her canoe at 3m/s. She heads downstreamat an angle of 30 degrees to the current which is flowing at 4 m/s.
a) How far downstream does Mary travel in 10s?
b) What is the length of time required to cross the river ifits width is 150 m?
To solve these problems, we need to break down the motion into its horizontal and vertical components. Let's start by defining the given information:
Given:
- Mary's canoe speed: 3 m/s
- Angle between Mary's direction and the current: 30 degrees
- Current speed: 4 m/s
a) How far downstream does Mary travel in 10s?
To determine how far downstream Mary travels, we need to find the horizontal component of her velocity. We can calculate it using trigonometry.
The horizontal component of Mary's canoe velocity can be found by multiplying the total velocity (3 m/s) by the cosine of the angle between her direction and the current (cos 30°).
Horizontal component of velocity (Vx) = Total velocity (3 m/s) * cos 30°
Vx = 3 m/s * cos 30°
Vx = 3 m/s * 0.866
Vx = 2.598 m/s
Now that we know the horizontal component of Mary's velocity, we can calculate the distance she travels in 10 seconds using the formula:
Distance = Velocity * Time
Distance = 2.598 m/s * 10 s
Distance = 25.98 m
Therefore, Mary travels approximately 25.98 meters downstream in 10 seconds.
b) What is the length of time required to cross the river if its width is 150 m?
To find the time it takes for Mary to cross the river, we need to determine the sum of her horizontal and vertical components of velocity.
The vertical component of velocity (Vy) can be found by multiplying the total velocity (3 m/s) by the sine of the angle between her direction and the current (sin 30°).
Vertical component of velocity (Vy) = Total velocity (3 m/s) * sin 30°
Vy = 3 m/s * sin 30°
Vy = 3 m/s * 0.5
Vy = 1.5 m/s
Now that we have both the horizontal (Vx) and vertical (Vy) components of velocity, we can use the Pythagorean theorem to find the resultant velocity (V).
V^2 = Vx^2 + Vy^2
V^2 = (2.598 m/s)^2 + (1.5 m/s)^2
V^2 = 6.744m^2/s^2 + 2.25m^2/s^2
V^2 = 8.994m^2/s^2
Taking the square root of both sides, we get:
V ≈ 2.999 m/s
Now that we know the resultant velocity, we can calculate the time it takes for Mary to cross the river using the formula:
Time = Distance / Velocity
Time = 150 m / 2.999 m/s
Time ≈ 50.02 s
Therefore, it would take approximately 50.02 seconds for Mary to cross the river, assuming the width is 150 meters.