A particle starts from rest accelerates at `2 m//s^2` for `10 s` and then goes for constant speed for `30 s` and then decelerates at `4 m//s^2` till it stops. What is the distance travelled by it.

Options

A. 750 m
B. 800 m
C. 700 m
D. 850 m

Explanation

To solve the problem of the particle's motion, we will break it down into three distinct stages: acceleration, constant speed, and deceleration. We will calculate the distance traveled in each stage and then sum them up to find the total distance.

### Step 1: Calculate Distance During Acceleration (s1)

The particle starts from rest and accelerates at \(2 \, \text{m/s}^2\) for \(10 \, \text{s}\).

Using the formula for distance under constant acceleration:
\[ s_1 = \frac{1}{2} a t^2 \]
where:
- \(a = 2 \, \text{m/s}^2\)
- \(t = 10 \, \text{s}\)

Substituting the values:
\[ s_1 = \frac{1}{2} \times 2 \times (10)^2 = \frac{1}{2} \times 2 \times 100 = 100 \, \text{m} \]

### Step 2: Calculate Distance During Constant Speed (s2)

After accelerating, the particle moves at a constant speed for \(30 \, \text{s}\). First, we need to find the speed at the end of the acceleration phase.

Using the formula for final velocity:
\[ v = u + at \]
where:
- \(u = 0 \, \text{m/s}\) (initial speed)
- \(a = 2 \, \text{m/s}^2\)
- \(t = 10 \, \text{s}\)

Substituting the values:
\[ v = 0 + 2 \times 10 = 20 \, \text{m/s} \]

Now, we can calculate the distance traveled at constant speed:
\[ s_2 = v \times t = 20 \, \text{m/s} \times 30 \, \text{s} = 600 \, \text{m} \]

### Step 3: Calculate Distance During Deceleration (s3)

The particle decelerates at \(4 \, \text{m/s}^2\) until it stops. We will use the third equation of motion to find the distance during deceleration:
\[ v^2 = u^2 + 2as \]
Here, the final velocity \(v = 0\) (since it stops), the initial velocity \(u = 20 \, \text{m/s}\), and \(a = -4 \, \text{m/s}^2\) (negative because it is deceleration).

Rearranging the equation to solve for \(s\):
\[ 0 = (20)^2 + 2 \times (-4) \times s \]
\[ 0 = 400 - 8s \]
\[ 8s = 400 \]
\[ s_3 = \frac{400}{8} = 50 \, \text{m} \]

### Step 4: Calculate Total Distance

Now, we can find the total distance traveled by the particle by summing the distances from all three stages:
\[ s = s_1 + s_2 + s_3 = 100 \, \text{m} + 600 \, \text{m} + 50 \, \text{m} = 750 \, \text{m} \]

### Final Answer

The total distance traveled by the particle is:
\[ \boxed{750 \, \text{m}} \]

---