We are given the following information in the question:

• A particle $P$ has a mass of 2 kg.
• There are two forces acting on $P$: $\mathbf{F_1} = (2\mathbf{i} + 4\mathbf{j}) \, \text{N} \quad \text{and} \quad \mathbf{F_2} = (c\mathbf{i} - 2\mathbf{j}) \, \text{N}$ where $c$ is a constant.
• The magnitude of the acceleration of $P$ is $\sqrt{5} \, \text{ms}^{-2}$.

We need to find the two possible values of $c$.

Step 1: Calculate the net force

The net force $\mathbf{F}$ acting on the particle is the sum of $\mathbf{F_1}$ and $\mathbf{F_2}$: $\mathbf{F} = \mathbf{F_1} + \mathbf{F_2} = (2 + c)\mathbf{i} + (4 - 2)\mathbf{j}$ This simplifies to: $\mathbf{F} = (2 + c)\mathbf{i} + 2\mathbf{j}$

Step 2: Use Newton's second law

Newton's second law states: $\mathbf{F} = m \mathbf{a}$ where $m = 2 \, \text{kg}$ is the mass and $\mathbf{a}$ is the acceleration.

Since the magnitude of the acceleration is given as $|\mathbf{a}| = \sqrt{5} \, \text{ms}^{-2}$, we can use this in our calculations. The magnitude of the net force is: $|\mathbf{F}| = m |\mathbf{a}| = 2 \times \sqrt{5} = 2\sqrt{5} \, \text{N}$

Step 3: Find the magnitude of the net force

The magnitude of the net force $\mathbf{F}$ is: $|\mathbf{F}| = \sqrt{(2 + c)^2 + 2^2}$ Simplifying the expression inside the square root: $|\mathbf{F}| = \sqrt{(2 + c)^2 + 4}$

Step 4: Set up the equation

Now, equate the magnitudes of the net force: $\sqrt{(2 + c)^2 + 4} = 2\sqrt{5}$ Square both sides to remove the square root: $(2 + c)^2 + 4 = 20$ Simplify: $(2 + c)^2 = 16$

Step 5: Solve for $c$

Take the square root of both sides: $2 + c = \pm 4$ Thus, the two possible values of $c$ are: $c = 4 - 2 = 2 \quad \text{or} \quad c = -4 - 2 = -6$

Final Answer:

The two possible values of $c$ are $c = 2$ and $c = -6$.

Do you have any questions, or would you like further clarification on any part of this solution?

Here are 5 related questions to deepen your understanding:

1. How does Newton's second law relate force and acceleration in general?
2. Why do we square both sides of the equation when solving for $c$?
3. Can you explain why vector components are added separately (i.e., $\mathbf{i}$ and $\mathbf{j}$ components)?
4. What happens to the motion of the particle if only one of the forces was applied?
5. How would the values of $c$ change if the mass of the particle were different?

Tip: When solving vector problems, always keep track of each component separately and treat them as independent dimensions.