A truck accelerates from speed v to 2v. Work done in during this is

Options

A. three times as the work done in accelerating it from rest to v
B. same as the work dine in accelerating it from rest to v
C. four times as the work done in accelerating it from rest to v
D. less than the work done in accelerating it from rest to v

Explanation

To solve the problem of work done when a truck accelerates from speed \( v \) to \( 2v \), we can use the concept of kinetic energy. The work done on the truck can be calculated using the change in its kinetic energy.

### Step-by-Step Solution:

1. **Understand Kinetic Energy**:
The kinetic energy (KE) of an object is given by the formula:
\[ KE = \frac{1}{2} mv^2 \]
where \( m \) is the mass of the object and \( v \) is its velocity.

2. **Calculate Initial Kinetic Energy**:
When the truck is moving at speed \( v \), its kinetic energy \( KE_1 \) is:
\[ KE_1 = \frac{1}{2} mv^2 \]

3. **Calculate Final Kinetic Energy**:
When the truck accelerates to speed \( 2v \), its kinetic energy \( KE_2 \) is:
\[ KE_2 = \frac{1}{2} m(2v)^2 = \frac{1}{2} m(4v^2) = 2mv^2 \]

4. **Determine the Change in Kinetic Energy**:
The work done \( W \) on the truck during the acceleration from speed \( v \) to \( 2v \) is equal to the change in kinetic energy:
\[ W = KE_2 - KE_1 \]
Substituting the values we calculated:
\[ W = 2mv^2 - \frac{1}{2} mv^2 \]
\[ W = 2mv^2 - 0.5mv^2 = \frac{4mv^2}{2} - \frac{1mv^2}{2} = \frac{3mv^2}{2} \]

5. **Compare with Work Done from Rest to Speed \( v \)**:
The work done \( W_0 \) to accelerate the truck from rest to speed \( v \) is:
\[ W_0 = \frac{1}{2} mv^2 \]

6. **Relate Work Done**:
Now, we can express the work done from \( v \) to \( 2v \) in terms of \( W_0 \):
\[ W = 3 \times W_0 \]
where \( W_0 = \frac{1}{2} mv^2 \).

7. **Conclusion**:
Therefore, the work done in accelerating the truck from speed \( v \) to \( 2v \) is three times the work done in accelerating it from rest to speed \( v \).

### Final Answer:
The work done in accelerating the truck from speed \( v \) to \( 2v \) is **three times the work done in accelerating it from rest to speed \( v \)**.

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