Arace car starts from rest and travels east along a straight and level track. for the first 5.0 s of the car's motion, the eastward component of the car's velocity is given by υx(t)=(0.980m/s3)t2.
what is the acceleration of the car when υx = 14.5 m/s ?

This question is dealing with velocity, acceleration and time of motion.

Acceleration is; aₓ = 7.54 m/s²

We are told that the eastward component of the car's velocity is;

uₓ(t) = (0.980 m/s³) t²

Now, from calculus differentiation in maths, we know that with respect to time, the derivative of velocity is equal to the acceleration.

Thus;

aₓ(t) = du/dt = 2t(0.980 m/s³)  

aₓ(t) = 1.96t m/s³

We w ant to find the acceleration of the car when velocity is; uₓ = 14.5 m/s. Let us find the time first and then plug the value into the acceleration equation.

Thus;

14.5 m/s = (0.980 m/s³) t²

14.5/0.98 = t²

t = 3.85 s

Putting 3.85 for t in the acceleration equation to get;

aₓ = (1.96 m/s³) (3.85 s)

aₓ = 7.54 m/s²

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7.54 m/s²

Explanation:

uₓ(t) = (0.980 m/s³) t²

Acceleration is the derivative of velocity with respect to time.

aₓ(t) = 2 (0.980 m/s³) t

aₓ(t) = (1.96 m/s³) t

When uₓ = 14.5 m/s, the time is:

14.5 m/s = (0.980 m/s³) t²

t = 3.85 s

Plugging into acceleration equation:

aₓ = (1.96 m/s³) (3.85 s)

aₓ = 7.54 m/s²

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