The minimum work done required to accelerate a truck on a horizontal road from rest to speed `v`

Options

A. is less than that required to accelerate it from `v` to `2v`.
B. is equal than that required to accelerate it from `v` to `2v`.
C. is more than that required to accelerate it from `v` to `2v`.
D. may be any one of the above since it depends on the force on the truck and the distance over which it acts.

Explanation

To find the minimum work done required to accelerate a truck from rest to speed \( v \) on a horizontal road, we can use the work-energy theorem. The work-energy theorem states that the work done on an object is equal to the change in its kinetic energy.

### Step-by-Step Solution:

1. **Understanding the Initial and Final States:**
- The truck starts from rest, which means its initial velocity \( u = 0 \).
- The final velocity after acceleration is \( v \).

2. **Calculating Initial and Final Kinetic Energy:**
- The initial kinetic energy \( KE_i \) when the truck is at rest is given by:
\[ KE_i = \frac{1}{2} m u^2 = \frac{1}{2} m (0)^2 = 0 \]
- The final kinetic energy \( KE_f \) when the truck reaches speed \( v \) is:
\[ KE_f = \frac{1}{2} m v^2 \]

3. **Finding the Change in Kinetic Energy:**
- The change in kinetic energy \( \Delta KE \) is:
\[ \Delta KE = KE_f - KE_i = \frac{1}{2} m v^2 - 0 = \frac{1}{2} m v^2 \]

4. **Applying the Work-Energy Theorem:**
- According to the work-energy theorem, the work done \( W \) on the truck to accelerate it from rest to speed \( v \) is equal to the change in kinetic energy:
\[ W = \Delta KE = \frac{1}{2} m v^2 \]

5. **Conclusion:**
- Therefore, the minimum work done required to accelerate the truck from rest to speed \( v \) is:
\[ W = \frac{1}{2} m v^2 \]