The velocity w of a particle in a time t is given by the expression \( \mathrm{W}=109+3 \mathrm{t}^{3} \). Find the instantaneous acceleration after 2 seconds

To find the instantaneous acceleration, you need to differentiate the velocity function with respect to time. Given the velocity function \( \mathrm{W} = 109 + 3\mathrm{t}^{3} \), the acceleration \( a \) is the derivative \( \frac{dW}{dt} \). Differentiating: \[ a = \frac{dW}{dt} = \frac{d}{dt}(109 + 3\mathrm{t}^{3}) = 0 + 9\mathrm{t}^{2} \] Now, plug in \( t = 2 \) seconds: \[ a(2) = 9(2^{2}) = 9 \times 4 = 36 \, \text{m/s}^{2} \] So, the instantaneous acceleration after 2 seconds is \( 36 \, \text{m/s}^{2} \). Acceleration can often feel mysterious, but it's essentially how quickly your velocity changes. Think of it like speeding up in a car; if you press the gas pedal, your speed increases - that’s acceleration in action!