To find the instantaneous acceleration of a particle given its velocity as a function of time, we need to follow a few steps. Let's break it down step-by-step. ### Step 1: Understand the Given Equation The velocity \( y \) of the particle is given by the equation: \[ y = 10 + 2t^2 \] Here, \( y \) represents the velocity in meters per second (ms\(^{-1}\)), and \( t \) represents time in seconds (s). ### Step 2: Find the Acceleration Acceleration is defined as the rate of change of velocity with respect to time. Mathematically, this is expressed as: \[ a = \frac{dy}{dt} \] To find the instantaneous acceleration, we need to differentiate the velocity function \( y \) with respect to time \( t \). ### Step 3: Differentiate the Velocity Function We will differentiate the equation \( y = 10 + 2t^2 \): - The derivative of a constant (10) is 0. - The derivative of \( 2t^2 \) is \( 4t \) (using the power rule, where the derivative of \( t^n \) is \( nt^{n-1} \)). Thus, the acceleration \( a \) is: \[ a = \frac{dy}{dt} = 0 + 4t = 4t \] ### Step 4: Calculate the Instantaneous Acceleration at \( t = 5 \) seconds Now, we substitute \( t = 5 \) seconds into the acceleration equation: \[ a = 4t = 4 \times 5 = 20 \, \text{ms}^{-2} \] ### Conclusion The instantaneous acceleration after 5 seconds is: \[ \boxed{20 \, \text{ms}^{-2}} \] ### Explanation of Options - **Option A: 60 ms\(^{-2}\)** - This is incorrect because it does not follow from the differentiation of the velocity function. It may arise from a misunderstanding of the relationship between velocity and acceleration. - **Option B: 20 ms\(^{-2}\)** - This is the correct answer, as we derived it directly from the velocity function by differentiating and substituting the correct time value. - **Option C: 15 ms\(^{-2}\)** - This option is incorrect. It may result from an incorrect calculation or misunderstanding of the differentiation process. - **Option D: 10 ms\(^{-2}\)** - This is also incorrect. It does not correspond to any calculation derived from the given velocity function. ### Common Pitfalls - **Misunderstanding the relationship between velocity and acceleration**: Remember that acceleration is the derivative of velocity, not the velocity itself. - **Forgetting to apply the power rule correctly**: Ensure you apply the differentiation rules accurately, especially for polynomial functions. - **Not substituting the correct time value**: Always double-check that you are substituting the correct time into your final acceleration equation. ### Revision Summary - Acceleration is the derivative of velocity with respect to time. - Differentiate the velocity function to find acceleration. - Substitute the given time into the acceleration equation to find instantaneous acceleration. - The correct answer for the instantaneous acceleration after 5 seconds is 20 ms\(^{-2}\).