A canoe of mass 70 kg with a paddler of mass 55 kg are in a river. If the river's current exerts a force of 15 N [E] while the paddler is paddling with an average force of 22 N [N38degreesW], find the acceleration of the canoe and paddler. Use both a component and trigonometric method
To find the acceleration of the canoe and paddler, we need to consider the forces acting on the system and use Newton's second law of motion.
1. Component method:
Let's break down the forces into their horizontal and vertical components:
The force of the river's current (15 N [E]) can be decomposed into its horizontal component and vertical component as follows:
- Horizontal component: 15 N * cos(90°) = 0 N [E]
- Vertical component: 15 N * sin(90°) = 15 N [N]
The force applied by the paddler (22 N [N38°W]) can also be decomposed into its horizontal and vertical components:
- Horizontal component: 22 N * cos(38°) = 17.19 N [W]
- Vertical component: 22 N * sin(38°) = 13.41 N [N]
Now, let's find the net horizontal and vertical forces acting on the canoe and paddler:
- Horizontal force: net force = 17.19 N [W] + 0 N [E] = 17.19 N [W]
- Vertical force: net force = 13.41 N [N] + 15 N [N] = 28.41 N [N]
Using Newton's second law, we can compute the acceleration in each direction:
- Horizontal acceleration: a_horizontal = net horizontal force / total mass = 17.19 N [W] / (70 kg + 55 kg) = 0.15 m/s² [W]
- Vertical acceleration: a_vertical = net vertical force / total mass = 28.41 N [N] / (70 kg + 55 kg) = 0.25 m/s² [N]
2. Trigonometric method:
Alternatively, we can use trigonometry to directly find the net force and angle of the net force:
Let θ be the angle between the net force and the horizontal axis.
To find θ:
tan(θ) = (13.41 N [N] + 15 N [N]) / 17.19 N [W]
θ ≈ arctan((13.41 N + 15 N) / 17.19 N) ≈ 59.8°
The magnitude of the net force can be found using the Pythagorean theorem:
net force = √((17.19 N [W])² + (13.41 N [N] + 15 N [N])²) ≈ √(296.09 N² + 861.18 N²) ≈ √1157.27 N² ≈ 34.02 N
Using Newton's second law, we can compute the acceleration in each direction:
- Horizontal acceleration: a_horizontal = net force * cos(θ) / total mass = 34.02 N * cos(59.8°) / (70 kg + 55 kg) ≈ 0.15 m/s² [W]
- Vertical acceleration: a_vertical = net force * sin(θ) / total mass = 34.02 N * sin(59.8°) / (70 kg + 55 kg) ≈ 0.25 m/s² [N]
Therefore, the acceleration of the canoe and paddler is approximately 0.15 m/s² [W] horizontally and 0.25 m/s² [N] vertically.
To find the acceleration of the canoe and paddler, we need to determine the net force acting on them.
Using the component method, we consider the forces acting in the east-west (horizontal) and north-south (vertical) directions separately.
Horizontal Forces:
The force exerted by the river's current is in the east direction (positive). Therefore, the force exerted by the river's current is 15 N [E].
The force exerted by the paddler can be resolved into horizontal and vertical components using trigonometry.
Given that the force exerted by the paddler is 22 N [N38°W], we can find its horizontal component:
Horizontal component = 22 N * cos(38°)
Horizontal component ≈ 22 N * 0.795
Horizontal component ≈ 17.49 N [W]
The net horizontal force is the vector sum of the force exerted by the river's current and the horizontal component of the paddler's force:
Net horizontal force = 15 N [E] - 17.49 N [W]
Net horizontal force ≈ -2.49 N [W]
Vertical Forces:
The force exerted by the river's current has no vertical component because the current moves horizontally.
The vertical component of the paddler's force can be found using trigonometry:
Vertical component = 22 N * sin(38°)
Vertical component ≈ 22 N * 0.615
Vertical component ≈ 13.53 N [N]
The net vertical force is the vector sum of the vertical components of the forces exerted by the paddler and the river's current:
Net vertical force = 13.53 N [N]
Now that we have the net horizontal and vertical forces, we can find the acceleration using Newton's second law:
Net force = mass * acceleration
Net horizontal force = (mass of canoe + mass of paddler) * horizontal acceleration
Net vertical force = (mass of canoe + mass of paddler) * vertical acceleration
Let's substitute the values we know:
-2.49 N [W] = (70 kg + 55 kg) * horizontal acceleration
13.53 N [N] = (70 kg + 55 kg) * vertical acceleration
Simplifying these equations, we find:
Horizontal acceleration ≈ -0.024 m/s²
Vertical acceleration ≈ 0.104 m/s²
Therefore, the acceleration of the canoe and paddler using the component method is approximately:
Horizontal acceleration ≈ -0.024 m/s² [W]
Vertical acceleration ≈ 0.104 m/s² [N]
Now, let's use the trigonometric method to find the same result.
Using the trigonometric method, we consider the forces acting on the canoe and paddler as vectors.
The force exerted by the river's current is a vector of magnitude 15 N, pointing in the east direction (positive x-axis). We can represent it as 15 N [E].
The force exerted by the paddler is a vector of magnitude 22 N, angled 38 degrees west of north. We can represent it as 22 N [38°W].
To find the net force acting on the canoe and paddler, we need to add these vectors together:
Horizontal component of the net force = -15 N * cos(0°) + 22 N * cos(38°)
Horizontal component ≈ -15 N + 17.49 N
Horizontal component ≈ 2.49 N [W]
Vertical component of the net force = 15 N * sin(0°) + 22 N * sin(38°)
Vertical component ≈ 0 N + 13.53 N
Vertical component ≈ 13.53 N [N]
Now that we have the horizontal and vertical components of the net force, we can find the acceleration using Newton's second law:
Net force = mass * acceleration
Net horizontal force = (mass of canoe + mass of paddler) * horizontal acceleration
Net vertical force = (mass of canoe + mass of paddler) * vertical acceleration
Substituting the values we know:
Horizontal component ≈ 2.49 N [W] = (70 kg + 55 kg) * horizontal acceleration
Vertical component ≈ 13.53 N [N] = (70 kg + 55 kg) * vertical acceleration
Simplifying these equations, we find:
Horizontal acceleration ≈ -0.024 m/s²
Vertical acceleration ≈ 0.104 m/s²
Therefore, the acceleration of the canoe and paddler using the trigonometric method is approximately:
Horizontal acceleration ≈ -0.024 m/s² [W]
Vertical acceleration ≈ 0.104 m/s² [N]
Add the two forces as vectors, and then use
Fnet = M a
to solve for the vector a.
Fnet (the vector sum of the two forces) and a will be in the same direction.
In this case, M = 70 + 55 = 125 kg
I assume you know how to do vector addition by adding components.