Aproton moves along the x axis according to the equation x = 38 t + 14 t2, where x is in meters and t is in seconds. calculate (a) the average velocity of the proton during the first 3.0 s of its motion, (b) the instantaneous velocity of the proton at t = 3.0 s, and (c) the instantaneous acceleration of the proton at t = 3.0 s.

a) 240 m

b) 122 m/s

c) 28 m/s²

Explanation:

Given:

Equation for motion

x = 38t + 14t²

a) average velocity during first 3 seconds

average velocity =

now,

distance, at t = 0 s

x = 38 × 0 + 14 × 0² = 0 m

distance, at t = 3 s

x = 38 × 3 + 14 × 3² = 240 m

therefore,

average velocity =  = 80 m/s

b) instantaneous velocity of the proton at t = 3.0 s

Instantaneous velocity, v =

or

Instantaneous velocity, v =

= 122 m/s

c) instantaneous acceleration of the proton at t = 3.0 s

Now,

Acceleration =  = 0 + 28 = 28 m/s²

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