To solve the problem of finding the displacement of a particle moving with an initial velocity and constant acceleration, we can break it down into two parts: first, we'll derive the formula for displacement in the last second, and then we'll plug in the given values to evaluate it.
Finding Displacement in the Last Second
When a particle moves with an initial velocity \( u \) and constant acceleration \( a \), the total displacement \( s \) over a time \( t \) can be calculated using the formula:
s = ut + \frac{1}{2} a t^2
However, to find the displacement during the last second, we need to consider the displacement from time \( t-1 \) to \( t \). We can calculate the displacement at time \( t \) and then subtract the displacement at time \( t-1 \).
Step-by-Step Calculation
1. **Displacement at time \( t \)**:
Using the formula:
s(t) = u t + \frac{1}{2} a t^2
2. **Displacement at time \( t-1 \)**:
Similarly, we calculate:
s(t-1) = u(t-1) + \frac{1}{2} a (t-1)^2
3. **Displacement in the last second**:
Now, we find the displacement during the last second by subtracting:
Displacement in last second = s(t) - s(t-1)
Substituting the expressions we derived:
Displacement in last second = [u t + \frac{1}{2} a t^2] - [u(t-1) + \frac{1}{2} a (t-1)^2]
4. **Simplifying the expression**:
After simplification, we find:
Displacement in last second = u + a(t - 0.5)
Evaluating with Given Values
Now, let's evaluate this expression using the provided values: \( u = 5 \, \text{m/s} \), \( a = 2 \, \text{m/s}^2 \), and \( t = 10 \, \text{s} \).
Substituting these values into the formula:
Displacement in last second = 5 + 2(10 - 0.5)
Calculating further:
Displacement in last second = 5 + 2(9.5)
Displacement in last second = 5 + 19 = 24 \, \text{m}
Final Result
Thus, the displacement of the particle during the last second of its motion is 24 meters.
