Q: If the position of a particle is given by $$x(t) = 4t^2$ + 2t$$, what is its instantaneous velocity at $$t = 3 s$$?

$$\frac{dx}{dt} = 8t + 2$$, so at $$t=3$$, $$v = 8 \times 3 + 2 = 26 m/s$

Q: A car accelerates from rest at $$2 m/s$^2$$ for $$5$$ seconds. What is its final velocity?

$$v = v_0$ + at = 0 + 2 \times 5 = 10 m/s$

Question 1

A car travels 120 m east and then 300 m north. What is the magnitude of its displacement?

Accepted Answer

$$\sqrt{120^2$ + 300^2$} = 323 m$

Question 2

Which of the following best describes the difference between distance and displacement?

Accepted Answer

Distance is the total length of the path traveled, while displacement is the straight-line vector from start to finish.

Question 3

A runner completes one lap around a 400 m circular track in 80 s. What is the runner's average velocity?

Accepted Answer

$$0 m/s$

Question 4

If the position of a particle is given by $$x(t) = 4t^2$ + 2t$$, what is its instantaneous velocity at $$t = 3 s$$?

Accepted Answer

$$\frac{dx}{dt} = 8t + 2$$, so at $$t=3$$, $$v = 8 	imes 3 + 2 = 26 m/s$

Question 5

A car accelerates from rest at $$2 m/s$^2$$ for $$5$$ seconds. What is its final velocity?

Accepted Answer

$$v = v_0$ + at = 0 + 2 	imes 5 = 10 m/s$

Question 6

Which of the following statements about frames of reference is correct?

Accepted Answer

A measured quantity can be different in different frames of reference.

Question 7

A boat moves north at $$10 m/s $$relative to the water. The river flows east at $$3 m/s. $$What is the boat's speed relative to the ground?

Accepted Answer

$$\sqrt{10^2$ + 3^2$} = 10.44 m/s$

Question 8

The area under a velocity-time ($$v vs$$. $$t) $$graph represents:

Accepted Answer

Displacement

Question 9

A truck moves at $$100 m/s $$east relative to the ground. A car moves at $$150 m/s $$east relative to the ground. What is the velocity of the truck relative to the car?

Accepted Answer

$$v_{truck,car} = v_{truck,ground} - v_{car,ground} = 100 - 150 = -50 m/s$

Question 10

Which of the following is the correct expression for average velocity?

Accepted Answer

$$\vec{v}_{avg} = \frac{\Delta \vec{r}}{\Delta t}$$

Question 11

A cyclist travels 60 m north, then 80 m east. What is the magnitude of the displacement?

Accepted Answer

$$\sqrt{60^2$ + 80^2$} = 100 m$

Question 12

If the velocity of an object is constant, what can be said about its acceleration?

Accepted Answer

Acceleration is zero.

Question 13

A car moves with a velocity of $$20 m/s $$for $$10 s$$, then stops in $$5 s $$with constant acceleration. What is the magnitude of the acceleration during stopping?

Accepted Answer

$$a = \frac{0 - 20}{5} = -4 m/s$^2 ($$magnitude is $$4 m/s$^2)$$

Question 14

Which of the following is true about instantaneous velocity?

Accepted Answer

It is the derivative of position with respect to time.

Question 15

A bus moves at $$5 m/s $$east relative to the ground. A passenger walks at $$1 m/s $$east relative to the bus. What is the passenger's velocity relative to the ground?

Accepted Answer

$$v_{passenger,ground} = v_{passenger,bus} + v_{bus,ground} = 1 + 5 = 6 m/s$

Question 16

The slope of a position-time ($$r vs$$. $$t) $$graph at any point gives:

Accepted Answer

Instantaneous velocity

Question 17

A car travels 100 km at 50 km/h and then another 100 km at 100 km/h. What is the average speed for the entire trip?

Accepted Answer

$$	ext{Total time} = \frac{100}{50} + \frac{100}{100} = 2 + 1 = 3 h$$; 	ext{Average speed} = \frac{200}{3} = 66.7$$ km/h$$

Question 18

Which of the following is the correct formula for the magnitude of a vector $$\vec{r} = x \hat{i} + y \hat{j}$$?

Accepted Answer

$$r = \sqrt{x^2$ + y^2$}$$

Question 19

A train moves at $$30 m/s $$east. A person walks at $$2 m/s $$west inside the train. What is the person's velocity relative to the ground?

Accepted Answer

$$v_{person,ground} = 30 + (-2) = 28 m/s$

Kinematics and Relative Motion: Study Notes for Physics with Calculus

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