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The problem asks for the velocity of a toy after 2 seconds, given the acceleration \( a(t) = 6t + 15 \) and an initial velocity of 2 m/s.

### Step-by-Step Solution:

1. **Given acceleration function:**
   \[
   a(t) = 6t + 15
   \]

2. **Find velocity function by integrating acceleration:**
   Velocity is the integral of acceleration:
   \[
   v(t) = \int a(t) \, dt = \int (6t + 15) \, dt
   \]
   Performing the integration:
   \[
   v(t) = 3t^2 + 15t + C
   \]
   where \( C \) is the constant of integration.

3. **Use initial condition to find \( C \):**
   The initial velocity \( v(0) = 2 \, \text{m/s} \):
   \[
   v(0) = 3(0)^2 + 15(0) + C = 2
   \]
   Thus, \( C = 2 \).

   So, the velocity function is:
   \[
   v(t) = 3t^2 + 15t + 2
   \]

4. **Find the velocity after 2 seconds:**
   Substitute \( t = 2 \) into the velocity function:
   \[
   v(2) = 3(2)^2 + 15(2) + 2
   \]
   Simplifying:
   \[
   v(2) = 3(4) + 30 + 2 = 12 + 30 + 2 = 44 \, \text{m/s}
   \]

### Final Answer:
The velocity after 2 seconds is \( 44 \, \text{m/s} \).

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Would you like further details or have any questions?

Here are 5 related questions for practice:

1. How would the result change if the initial velocity was 0 m/s instead of 2 m/s?
2. What is the velocity after 4 seconds?
3. Find the position function given that the position at \( t = 0 \) is 0.
4. How does the velocity change over time, based on the velocity function?
5. How would you find the time at which the toy reaches a specific velocity, say 60 m/s?

**Tip:** When solving kinematic problems, always check if you need to integrate or differentiate based on whether you're given acceleration, velocity, or position!

Find the velocity after 2 seconds if it has an initial velocity of 2 m/s, given that the acceleration a(t) = 6t + 15.

Solve a kinematics problem involving the acceleration function a(t) = 6t + 15 to find the velocity of a toy after 2 seconds, with an initial velocity of 2 m/s. Learn how to apply calculus concepts like integration and the fundamental theorem of calculus to solve this physics problem.

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