9.1. Laplace Transform of Unit Step Function

Try these first

1. 1.

   Define the unit step function.

   Hint

   Think about how the function behaves over time.

2. 2.

   What is the Laplace Transform of u(t−a)?

   Hint

   Recall the formula we derived.

3. 3.

   What is the definition of the unit step function?

   Hint

   Consider how the function activates.

4. 4.

   The Laplace Transform of u(t−a) is given by which formula?

   • e^{−as}/s
   • s/(e^{as})
   • (1/s)e^{−as}

   Hint

   Recall the transformation formula we discussed.

5. 5.

   Consider a switch that activates at t = 3 in a system described by the function f(t) = cos(t). Calculate the Laplace Transform of the system taking into account the unit step function.

   Hint

   Use the property of shifting and recognize cos(t) transforms.

6. 6.

   A mechanical system experiences a sudden force at t = 4 described by the impulse m*d²x/dt² + kx = F(t)u(t−4). Derive the system response in the Laplace domain.

   Hint

   Focus on the relationship established through the Laplace properties of derivatives.