Let's solve the problem step-by-step.
First, we need to understand the given question:
A car of mass 1500 kg is accelerating from rest to a speed of 20 m/s in 6.0 seconds. We need to find the average power produced by the moving car.
We'll break this down into smaller steps:
### Step 1: Calculate Acceleration
The car starts from rest (initial velocity [tex]\(u = 0\)[/tex]) and reaches a final velocity ([tex]\(v = 20 \, \text{m/s}\)[/tex]) over a time period ([tex]\(t = 6.0 \, \text{seconds}\)[/tex]). The formula for acceleration ([tex]\(a\)[/tex]) is given by:
[tex]\[ \text{Acceleration} (a) = \frac{v - u}{t} \][/tex]
Putting in the values:
[tex]\[ a = \frac{20 \, \text{m/s} - 0 \, \text{m/s}}{6.0 \, \text{s}} = \frac{20}{6} = 3.333 \, \text{m/s}^2 \][/tex]
### Step 2: Calculate Force
Force ([tex]\(F\)[/tex]) is given by Newton's second law of motion ([tex]\(F = m \cdot a\)[/tex]), where [tex]\(m\)[/tex] is the mass of the car.
[tex]\[ F = 1500 \, \text{kg} \times 3.333 \, \text{m/s}^2 = 5000 \, \text{N} \][/tex]
### Step 3: Calculate Distance Travelled
To calculate the distance ([tex]\(s\)[/tex]) travelled by the car, we use the kinematic equation:
[tex]\[ s = ut + \frac{1}{2} a t^2 \][/tex]
Since the car starts from rest ([tex]\(u = 0\)[/tex]):
[tex]\[ s = 0 + \frac{1}{2} \times 3.333 \, \text{m/s}^2 \times (6 \, \text{s})^2 \][/tex]
[tex]\[ s = 0.5 \times 3.333 \times 36 \][/tex]
[tex]\[ s = 60 \, \text{m} \][/tex]
### Step 4: Calculate Work Done
Work done ([tex]\(W\)[/tex]) is given by the formula:
[tex]\[ W = F \times s \][/tex]
[tex]\[ W = 5000 \, \text{N} \times 60 \, \text{m} = 300000 \, \text{J} \][/tex]
### Step 5: Calculate Average Power
Power ([tex]\(P\)[/tex]) is the work done per unit time:
[tex]\[ P = \frac{W}{t} \][/tex]
[tex]\[ P = \frac{300000 \, \text{J}}{6 \, \text{s}} = 50000 \, \text{W} \][/tex]
### Final Answer
The average power produced by the moving car is [tex]\(50,000 \, \text{W}\)[/tex] or [tex]\(50 \, \text{kW}\)[/tex].
