The ramp function, denoted as 𝑢(𝑡)u(t), is defined as:a)𝑢(𝑡)=𝑡u(t)=t for 𝑡<0t<0 and 𝑢(𝑡)=1u(t)=1 for 𝑡≥0t≥0b)𝑢(𝑡)=1u(t)=1 for 𝑡<0t<0 and 𝑢(𝑡)=𝑡u(t)=t for 𝑡≥0t≥0c)𝑢(𝑡)=1u(t)=1 for 𝑡<0t<0 and 𝑢(𝑡)=0u(t)=0 for 𝑡≥0t≥0d)𝑢(𝑡)=0u(t)=0 for 𝑡<0t<0 and 𝑢(𝑡)=1u(t)=1 for 𝑡≥0t≥0

The Laplace transform of a unit-ramp function starting at t = a is

Express the functionf(t)=⎧⎩⎨⎪⎪−7sin(6t),2−9t2,0,0≤t<66≤t<8otherwise𝑓(𝑡)={−7sin⁡(6𝑡),0≤𝑡<62−9𝑡2,6≤𝑡<80,𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒using Heaviside functions.Note: You must use the notation u(t−c)𝑢(𝑡−𝑐) rather than uc(t)𝑢𝑐(𝑡) in order for your answer to be accepted by Bolster Academy.

Express the functionf(t)={4cos(9t),0,7≤t<8otherwise𝑓(𝑡)={4cos⁡(9𝑡),7≤𝑡<80,𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒using Heaviside functions.Note: You must use the notation u(t−c)𝑢(𝑡−𝑐) rather than uc(t)𝑢𝑐(𝑡) in order for your answer to be accepted by Bolster Academy.

For the situation described: (1) Simplify the work-energy equation to its proper form. (2) Then identify the letter of the corresponding work-energy bar chart.A physics cart is pulled from the bottom of a ramp (initial state) to the top of the ramp (final state) at a constant speed.

(b) Sketch the graph of displacement against distance graph for progressive wave above ina range of 0 ≤ 𝑥 ≤ 𝜆 at time, t = 0 s.