Q: If the position of a particle is given by $$x(t) = 3t^2$ + 2t + 1$$, what is its instantaneous velocity at $$t = 2\ \text{s}$$?

$$v = \dfrac{dx}{dt} = 6t + 2 \Rightarrow v(2) = 14\ \text{m/s}$$

Q: A ball is tossed upward from the ground and falls back down. At the highest point in its path, which statement is correct?

Velocity $$v = 0$$, but acceleration $$a \neq 0$$

Q: A car accelerates from rest with constant acceleration $$a = 2\ \text{m/s}^2. $$How far does it travel in 5 seconds?

$$x = \dfrac{1}{2} a t^2$ = \dfrac{1}{2} \times 2 \times 25 = 25\ \text{m}$$

Question 1

A car is initially traveling at $$v_0 = 29$$\ 	ext{m/s}$$ and comes to a stop by braking with a constant acceleration a_b = -9.81$$\ 	ext{m/s}^2$$. What is the stopping distance x_f$?

Accepted Answer

$$x_f = \dfrac{v_0$^2}{2a_b}$$

Question 2

Which of the following best describes the difference between velocity and speed?

Accepted Answer

Velocity is a vector quantity, while speed is a scalar quantity.

Question 3

A ball is thrown straight up and returns to its initial position. What is the average velocity and average acceleration during its flight?

Accepted Answer

Average velocity is zero, but average acceleration is not zero.

Question 4

If the position of a particle is given by $$x(t) = 3t^2$ + 2t + 1$$, what is its instantaneous velocity at $$t = 2\ 	ext{s}$$?

Accepted Answer

$$v = \dfrac{dx}{dt} = 6t + 2 \Rightarrow v(2) = 14\ 	ext{m/s}$$

Question 5

A ball is tossed upward from the ground and falls back down. At the highest point in its path, which statement is correct?

Accepted Answer

Velocity $$v = 0$$, but acceleration $$a 
eq 0$$

Question 6

Which equation gives the instantaneous velocity as a function of time for motion with constant acceleration?

Accepted Answer

$$v(t) = v_0$ + a t$$

Question 7

A particle moves along a straight line with position given by $$x(t) = 5t^3$ - 2t^2$ + 4. $$What is its acceleration at $$t = 1\ 	ext{s}$$?

Accepted Answer

$$a = \dfrac{d^2$x}{dt^2$} = 30t - 4 \Rightarrow a(1) = 26\ 	ext{m/s}^2$$

Question 8

A car accelerates from rest with constant acceleration $$a = 2\ 	ext{m/s}^2. $$How far does it travel in 5 seconds?

Accepted Answer

$$x = \dfrac{1}{2} a t^2$ = \dfrac{1}{2} 	imes 2 	imes 25 = 25\ 	ext{m}$$

Question 9

Which of the following is the correct calculus-based definition of instantaneous velocity?

Accepted Answer

$$v(t) = \dfrac{dx}{dt}$$

Question 10

A ball is dropped from a height of $$2\ 	ext{m}$$ above the ground. Which graph best describes its velocity $$v$$ versus time $$t$$?

Accepted Answer

A straight line with negative slope, starting at $$v=0$$ and decreasing linearly with time.

Question 11

If a particle’s velocity is given by $$v(t) = 4t - 3$$, what is its displacement between $$t = 0$$ and $$t = 2\ 	ext{s}$$?

Accepted Answer

$$\Delta x = \int_{0}^{2} (4t - 3) dt = [2t^2$ - 3t]_{0}^{2} = (8 - 6) - (0 - 0) = 2\ 	ext{m}$$

Question 12

Which formula is used to find the time $$t it $$takes for an object to stop when its initial velocity is $$v_0$$ and acceleration is $$a$$?

Accepted Answer

$$t = \dfrac{v_0$}{|a|}$$

Question 13

A particle moves with constant acceleration $$a. $$Which equation relates its displacement $$x to $$its initial velocity $$v_0$$, acceleration $$a$$, and time $$t$$?

Accepted Answer

$$x = x_0$ + v_0$ t + \dfrac{1}{2} a t^2$

Question 14

Which of the following statements about acceleration is correct?

Accepted Answer

Acceleration is the rate of change of velocity.

Question 15

A car travels 100 meters in 10 seconds. What is its average velocity if it started and ended at the same position?

Accepted Answer

Average velocity is $$0\ 	ext{m/s}$$

Question 16

Which equation is derived using the chain rule for constant acceleration?

Accepted Answer

$$v^2$$ = $$v_0^2 + 2a(x$$ - $$x_0$$)$$

Question 17

A runner accelerates from $$2\ 	ext{m/s} to$$ $$6\ 	ext{m/s} in$$ $$4\ 	ext{s}. $$What is the average acceleration?

Accepted Answer

$$a_{ave} = \dfrac{6 - 2}{4} = 1\ 	ext{m/s}^2$$

Question 18

If a particle’s position is $$x(t) = 10 - 4t$$, what is its velocity and acceleration?

Accepted Answer

Velocity $$v = -4\ 	ext{m/s}$$, acceleration $$a = 0$$

Question 19

Which of the following is NOT a valid step when solving a physics problem involving kinematics?

Accepted Answer

Ignoring units throughout the calculation.

Physics with Calculus: Motion Along a Straight Line (1-D Kinematics)

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