Motion (Kinematics) — Hard Practice Quiz

A Physics cheat sheet for Motion (Kinematics) — every key formula with its symbols defined — plus a hard-level practice quiz to test recall.

Formulas & key concepts

Final Velocity (Constant Acceleration): The final velocity \(v_f\) of an object is its initial velocity \(v_i\) plus the acceleration \(a\) multiplied by the time \(t\).

$$v_f = v_i + at$$

Displacement (Constant Acceleration): The displacement \(d\) of an object is its initial velocity \(v_i\) times time \(t\) plus half the acceleration \(a\) times the square of the time.

$$d = v_i t + \frac{1}{2}at^2$$

Velocity-Displacement Relationship: Relates the final velocity \(v_f\), initial velocity \(v_i\), constant acceleration \(a\), and displacement \(d\) without using time.

$$v_f^2 = v_i^2 + 2ad$$

Displacement with Average Velocity: The displacement \(d\) of an object is the average of its initial and final velocities multiplied by the time \(t\).

$$d = \frac{1}{2}(v_i + v_f)t$$

Average Speed: The total distance traveled divided by the total time elapsed.

$$\text{Average Speed} = \frac{\text{Distance}}{\text{Elapsed Time}}$$

Relative Velocity: The velocity of object A relative to object C is the vector sum of the velocity of A relative to B and the velocity of B relative to C.

$$\vec{v}_{AC} = \vec{v}_{AB} + \vec{v}_{BC}$$

Projectile Motion (Horizontal Displacement): The horizontal position \(x\) at time \(t\), assuming constant horizontal velocity \(v_{0x}\).

$$x = v_{0x}t$$

Projectile Motion (Vertical Displacement): The vertical position \(y\) at time \(t\), with initial vertical velocity \(v_{0y}\) and vertical acceleration \(a_y\).

$$y = v_{0y}t + \frac{1}{2}a_yt^2$$

Projectile Motion (Vertical Velocity): The vertical velocity \(v_y\) at time \(t\), given initial vertical velocity \(v_{0y}\) and vertical acceleration \(a_y\).

$$v_y = v_{0y} + a_yt$$

Practice quiz

  1. An object starts from rest and accelerates uniformly. If its final velocity is $v$ after time $t$, what is its displacement if it accelerates for $2t$ with the same acceleration?

    • $2d$
    • $4d$
    • $d/2$
    • $d$

    Answer: $4d$

  2. A car accelerates from rest at $2 \text{ m/s}^2$ for $10 \text{ s}$. It then travels at a constant velocity for $20 \text{ s}$. What is the total distance covered?

    • $400 \text{ m}$
    • $500 \text{ m}$
    • $600 \text{ m}$
    • $300 \text{ m}$

    Answer: $500 \text{ m}$

  3. A projectile is launched horizontally from a height $H$ with an initial velocity $v_0$. If it lands at a horizontal distance $R$, what is the expression for $v_0$ in terms of $H$, $R$, and $g$ (acceleration due to gravity)?

    • $R\sqrt{\frac{g}{2H}}$
    • $R\sqrt{\frac{2g}{H}}$
    • $\frac{R}{t}$
    • $\frac{R^2 g}{2H}$

    Answer: $R\sqrt{\frac{g}{2H}}$

  4. A boat travels across a river with a velocity $\vec{v}_{BR}$ relative to the river, and the river flows with a velocity $\vec{v}_{RS}$ relative to the shore. If the boat aims directly across the river, and the river's speed doubles while the boat's speed relative to the river remains constant, how does the time to cross the river change?

    • It doubles.
    • It halves.
    • It remains unchanged.
    • It increases by a factor of $\sqrt{2}$.

    Answer: It remains unchanged.

  5. An object starts from rest and accelerates at $a_1$ for a time $t_1$. It then decelerates at $a_2$ until it comes to a stop. What is the total distance traveled?

    • $\frac{1}{2}a_1 t_1^2 (1 + \frac{a_1}{a_2})$
    • $\frac{1}{2}a_1 t_1^2 (1 - \frac{a_1}{a_2})$
    • $a_1 t_1^2 + \frac{a_1^2 t_1^2}{2a_2}$
    • $\frac{a_1 t_1^2}{2a_2}$

    Answer: $\frac{1}{2}a_1 t_1^2 (1 + \frac{a_1}{a_2})$

  6. A projectile is launched with an initial velocity $v_0$ at an angle $\theta$ above the horizontal. If the maximum height reached is $H$ and the horizontal range is $R$, what is the relationship between $H$ and $R$ if $\theta = 45^\circ$? (Assume $a_y = -g$)

    • $R = 2H$
    • $R = 4H$
    • $R = H$
    • $R = H/2$

    Answer: $R = 4H$

  7. Two cars, A and B, are initially $100 \text{ m}$ apart. Car A starts from rest and accelerates at $2 \text{ m/s}^2$ towards B. Car B starts simultaneously from rest and accelerates at $1 \text{ m/s}^2$ towards A. How long does it take for them to meet?

    • $10 \text{ s}$
    • $10\frac{\sqrt{6}}{3} \text{ s}$
    • $5 \text{ s}$
    • $20 \text{ s}$

    Answer: $10\frac{\sqrt{6}}{3} \text{ s}$

  8. A particle travels from point A to point B with a constant speed $v_1$ and returns from B to A with a constant speed $v_2$. What is the average speed for the entire round trip?

    • $\frac{v_1+v_2}{2}$
    • $\frac{2v_1 v_2}{v_1+v_2}$
    • $\sqrt{v_1 v_2}$
    • $\frac{v_1 v_2}{v_1+v_2}$

    Answer: $\frac{2v_1 v_2}{v_1+v_2}$

  9. An object starts from rest and undergoes constant acceleration $a$. If it covers a distance $d_1$ in the first $t$ seconds and a distance $d_2$ in the next $t$ seconds, what is the ratio $d_2/d_1$?

    • $1$
    • $2$
    • $3$
    • $4$

    Answer: $3$

  10. A ball is thrown vertically upwards with an initial velocity $v_0$. What is its velocity when it has reached half of its maximum height? (Assume acceleration due to gravity is $g$ downwards).

    • $\frac{v_0}{2}$
    • $\frac{v_0}{\sqrt{2}}$
    • $v_0\sqrt{2}$
    • $v_0$

    Answer: $\frac{v_0}{\sqrt{2}}$

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