Q: Given the position function $$x(t) = 4t^2$ - 2t + 1$$, what is the instantaneous velocity at $$t = 3\ \mathrm{s}$$?

$$v = \frac{dx}{dt} = 8t - 2$$, so\ v = 8 \times 3 - 2 = 22\ \mathrm{m/s}$

Q: A car accelerates from rest at $$a = 2\ \mathrm{m/s^2$}$$ for $$5\ \mathrm{s}. $$What is its final velocity?

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

Q: A projectile is launched at an angle $$\theta$$ with initial speed $$v_0. $$What is the formula for the range $$R on $$level ground?

$$R = \frac{v_0$^2 \sin 2\theta}{g}$$

Question 1

A car moves from position $$x_1 = 2$$\ \mathrm{m}$$ to x_2$ = 8\ \mathrm{m} in$$ $$3\ \mathrm{s}. $$What is the displacement vector $$\Delta x$$ and the average velocity $$v_{avg}$$?

Accepted Answer

$$\Delta x = 6\ \mathrm{m}$$,\ v_{avg} = 2\ \mathrm{m/s}$

Question 2

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

Accepted Answer

Displacement is a vector quantity, while distance is a scalar quantity.

Question 3

Given the position function $$x(t) = 4t^2$ - 2t + 1$$, what is the instantaneous velocity at $$t = 3\ \mathrm{s}$$?

Accepted Answer

$$v = \frac{dx}{dt} = 8t - 2$$, so\ v = 8 	imes 3 - 2 = 22\ \mathrm{m/s}$

Question 4

A runner moves in a straight line with a velocity given by $$v(t) = 3t^2$. What is the displacement between t = 0$ and t = 2$$\ \mathrm{s}$$?

Accepted Answer

$$\Delta x = \$$int_0$$^2 3t^2$ dt = [t^3$]_0^2$ = 8\ \mathrm{m}$$

Question 5

If a particle's velocity changes from $$10\ \mathrm{m/s} to$$ $$4\ \mathrm{m/s} in$$ $$3\ \mathrm{s}$$, what is its average acceleration?

Accepted Answer

$$a_{avg} = \frac{4 - 10}{3} = -2\ \mathrm{m/s^2$}$$

Question 6

A ball is thrown vertically upward with an initial velocity of $$20\ \mathrm{m/s}. $$What is the maximum height reached? (Assume $$g = 10\ \mathrm{m/s^2$})$$

Accepted Answer

$$h_{max} = \frac{v_0$^2}{2g} = \frac{400}{20} = 20\ \mathrm{m}$$

Question 7

Which of the following graphs best represents the velocity-time graph for an object moving with constant acceleration?

Accepted Answer

A straight line with a non-zero slope

Question 8

A car accelerates from rest at $$a = 2\ \mathrm{m/s^2$}$$ for $$5\ \mathrm{s}. $$What is its final velocity?

Accepted Answer

$$v = v_0$ + at = 0 + 2 	imes 5 = 10\ \mathrm{m/s}$$

Question 9

Given a velocity vector $$\vec{v} = 3\hat{i} + 4\hat{j}$$, what is the magnitude of the velocity?

Accepted Answer

$$|\vec{v}| = \sqrt{3^2$ + 4^2$} = 5$$

Question 10

A projectile is launched at an angle $$	heta$$ with initial speed $$v_0. $$What is the formula for the range $$R on $$level ground?

Accepted Answer

$$R = \frac{v_0$^2 \sin 2	heta}{g}$$

Question 11

If the position of a particle is given by $$\vec{r}(t) = (2t)\hat{i} + (3t^2$)\hat{j}$$, what is the velocity vector at $$t = 2\ \mathrm{s}$$?

Accepted Answer

$$\vec{v}(t) = \frac{d\vec{r}}{dt} = 2\hat{i} + 6t\hat{j}$$, so\ \vec{v}(2) = 2\hat{i} + 12\hat{j}$

Question 12

A cyclist travels 3 km east, then 4 km north. What is the magnitude of the displacement?

Accepted Answer

$$\sqrt{3^2$ + 4^2$} = 5\ \mathrm{km}$$

Question 13

Which of the following statements is true about acceleration?

Accepted Answer

Acceleration is the rate of change of velocity with respect to time.

Question 14

A car moves with a velocity $$v(t) = 5 - 2t (in m/s). At $$what time does the car stop?

Accepted Answer

$$v(t) = 0 \Rightarrow 5 - 2t = 0 \Rightarrow t = 2.5\ \mathrm{s}$$

Question 15

If a velocity-time graph is a straight line with a negative slope, what does this indicate about the object's motion?

Accepted Answer

The object is slowing down with constant acceleration.

Question 16

A ball is dropped from rest. How far does it fall in the first 2 seconds? (Assume $$g = 9.8\ \mathrm{m/s^2$})$$

Accepted Answer

$$s = \frac{1}{2}gt^2$ = \frac{1}{2} 	imes 9.8 	imes 4 = 19.6\ \mathrm{m}$$

Kinematics: Position, Displacement, Velocity, and Acceleration (Physics with Calculus)

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