Quiz & Worksheet - Linear Motion & Differentiation | Study.com - Free Printable

Let’s solve each question step by step, using the principles of calculus and kinematics:

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Question 1:

> The position of an object can be described by the function:
> x(t) = 12t - 7
> Find the velocity of this object at t = 2 s.

✔ Concept:
Velocity is the first derivative of position with respect to time:
v(t) = dx/dt

So, differentiate x(t):

> x(t) = 12t - 7
> v(t) = d/dt (12t - 7) = 12 m/s

This is a constant — velocity does not change with time.

Therefore, at t = 2 s, velocity is still 12 m/s.

✔ Answer: 12 m/s

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Question 2:

> The velocity of an object can be described by v(t) = 10 m/s.
> What is the acceleration of this object at t = 1 s?

✔ Concept:
Acceleration is the first derivative of velocity with respect to time:
a(t) = dv/dt

Here, v(t) = 10 → constant.

> a(t) = d/dt (10) = 0 m/s²

Since velocity is constant, acceleration is zero — no matter what time you pick.

✔ Answer: 0

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Question 3:

> Given the position function:
> x(t) = -t³ + 2t
> Find the acceleration of the particle at t = 3.0 s.

✔ Concept:
Acceleration is the second derivative of position with respect to time:
a(t) = d²x/dt²

Step 1: Find velocity (first derivative)

> x(t) = -t³ + 2t
> v(t) = dx/dt = -3t² + 2

Step 2: Find acceleration (second derivative)

> a(t) = dv/dt = d/dt (-3t² + 2) = -6t

Now plug in t = 3.0 s:

> a(3.0) = -6 × 3.0 = -18 m/s²

✔ Answer: -18 m/s²

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## ✔ Final Answers:

1. 12 m/s
2. 0
3. -18 m/s²

These are the correct choices based on differentiation rules and kinematic definitions.