A constant force of 2.50 N accelerates a stationary particle of mass 15 g through a displacement of 2.50 m. Find the work done and the average power delivered.

Explanation

To solve the problem, we will follow these steps:

### Step 1: Convert the mass from grams to kilograms
The mass of the particle is given as 15 g. Since the SI unit of mass is kilograms, we need to convert grams to kilograms.

\[ \text{Mass} = 15 \, \text{g} = \frac{15}{1000} \, \text{kg} = 0.015 \, \text{kg} \]

### Step 2: Calculate the work done by the force
The work done (W) by a constant force can be calculated using the formula:

\[ W = F \cdot d \]

where:
- \(W\) is the work done,
- \(F\) is the force applied (2.50 N),
- \(d\) is the displacement (2.50 m).

Substituting the values into the formula:

\[ W = 2.50 \, \text{N} \cdot 2.50 \, \text{m} = 6.25 \, \text{J} \]

### Step 3: Calculate the average power delivered
Average power (P) can be calculated using the formula:

\[ P = \frac{W}{t} \]

where:
- \(P\) is the average power,
- \(W\) is the work done (6.25 J),
- \(t\) is the time taken to do the work.

To find the time taken, we first need to find the acceleration (a) using Newton's second law:

\[ F = m \cdot a \implies a = \frac{F}{m} \]

Substituting the values:

\[ a = \frac{2.50 \, \text{N}}{0.015 \, \text{kg}} = 166.67 \, \text{m/s}^2 \]

Now, we can use the kinematic equation to find the time taken (t) for the displacement (d):

\[ d = \frac{1}{2} a t^2 \]

Rearranging to find \(t\):

\[ t^2 = \frac{2d}{a} \implies t = \sqrt{\frac{2d}{a}} \]

Substituting the values:

\[ t = \sqrt{\frac{2 \cdot 2.50 \, \text{m}}{166.67 \, \text{m/s}^2}} = \sqrt{\frac{5.00}{166.67}} \approx \sqrt{0.030} \approx 0.173 \, \text{s} \]

Now we can calculate the average power:

\[ P = \frac{6.25 \, \text{J}}{0.173 \, \text{s}} \approx 36.1 \, \text{W} \]

### Final Answers:
- Work done: \(6.25 \, \text{J}\)
- Average power delivered: \(36.1 \, \text{W}\)

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