Q: A soccer ball is kicked from the ground with an initial speed $v_0$ at an angle $\theta$ above the horizontal. Which equation gives the maximum height $h_{max}$ reached by the ball?

$h_{max} = \dfrac{v_0^2 \sin^2 \theta}{2g}$

Q: A stone is thrown from the top of a building with velocity $v_0$ at an angle $\theta$ above the horizontal. If the stone lands at a horizontal distance $R$ from the base, which equation gives $R$?

$R = v_0 \cos \theta \cdot t_{total}$

Q: A projectile is launched with initial velocity $v_0$ at angle $\theta$. What is the time of flight before it returns to the same vertical level?

$t = \dfrac{2 v_0 \sin \theta}{g}$

Q: A projectile is launched at $v_0$ and angle $\theta$. What is the formula for its vertical displacement $\Delta y$ after time $t$?

$\Delta y = v_0 \sin \theta \cdot t - \dfrac{1}{2} g t^2$

Q: A projectile is launched at $v_0$ and angle $\theta$. What is the formula for its range $R$ on level ground?

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

Q: A ball is thrown at $v_0$ and angle $\theta$. What is the formula for its velocity vector $\vec{v}$ at time $t$?

$\vec{v} = (v_0 \cos \theta) \hat{i} + (v_0 \sin \theta - g t) \hat{j}$

Q: A projectile is launched at $v_0$ and angle $\theta$. What is the formula for its speed $v$ at any time $t$?

$v = \sqrt{(v_0 \cos \theta)^2 + (v_0 \sin \theta - g t)^2}$

Q: A boat moves north at $4\,\mathrm{m/s}$ relative to the water, while the water flows east at $3\,\mathrm{m/s}$. What is the direction of the boat's velocity relative to the ground?

$\theta = \tan^{-1} \left( \dfrac{3}{4} \right)$ \, east of north

Question 1

A soccer ball is kicked from the ground with an initial speed $v_0$ at an angle $	heta$ above the horizontal. Which equation gives the maximum height $h_{max}$ reached by the ball?

Accepted Answer

$h_{max} = \dfrac{v_0^2 \sin^2 	heta}{2g}$

Question 2

A projectile is launched horizontally from a cliff of height $h$ with initial speed $v_0$. Neglecting air resistance, what is the time $t$ it takes to hit the ground?

Accepted Answer

$t = \sqrt{\dfrac{2h}{g}}$

Question 3

A ball is thrown with initial velocity $v_0$ at an angle $	heta$ above the horizontal. What is the horizontal component of its velocity?

Accepted Answer

$v_{0x} = v_0 \cos 	heta$

Question 4

A stone is thrown from the top of a building with velocity $v_0$ at an angle $	heta$ above the horizontal. If the stone lands at a horizontal distance $R$ from the base, which equation gives $R$?

Accepted Answer

$R = v_0 \cos 	heta \cdot t_{total}$

Question 5

A car moves east at $20\,\mathrm{m/s}$ and a train moves north at $30\,\mathrm{m/s}$. What is the magnitude of the car's velocity relative to the train?

Accepted Answer

$\sqrt{20^2 + 30^2} = 36.1\,\mathrm{m/s}$

Question 6

A projectile is launched with initial velocity $v_0$ at angle $	heta$. What is the time of flight before it returns to the same vertical level?

Accepted Answer

$t = \dfrac{2 v_0 \sin 	heta}{g}$

Question 7

A river flows east at $5\,\mathrm{m/s}$. A swimmer can swim north at $3\,\mathrm{m/s}$ relative to the water. What is the magnitude of the swimmer's velocity relative to the ground?

Accepted Answer

$\sqrt{5^2 + 3^2} = 5.83\,\mathrm{m/s}$

Question 8

A ball is thrown upward with initial velocity $v_0$. What is its velocity at the maximum height?

Accepted Answer

$v = 0$

Question 9

A projectile is launched at $v_0$ and angle $	heta$. What is the formula for its vertical displacement $\Delta y$ after time $t$?

Accepted Answer

$\Delta y = v_0 \sin 	heta \cdot t - \dfrac{1}{2} g t^2$

Question 10

A plane drops a package while flying horizontally at $v_0$. If the package lands a distance $d$ from the drop point, what is the time to hit the ground?

Accepted Answer

$t = \dfrac{d}{v_0}$

Question 11

A projectile is launched at $v_0$ and angle $	heta$. What is the formula for its range $R$ on level ground?

Accepted Answer

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

Question 12

A ball is thrown at $v_0$ and angle $	heta$. What is the formula for its velocity vector $\vec{v}$ at time $t$?

Accepted Answer

$\vec{v} = (v_0 \cos 	heta) \hat{i} + (v_0 \sin 	heta - g t) \hat{j}$

Question 13

A projectile is launched at $v_0$ and angle $	heta$. What is the formula for its speed $v$ at any time $t$?

Accepted Answer

$v = \sqrt{(v_0 \cos 	heta)^2 + (v_0 \sin 	heta - g t)^2}$

Question 14

A boat moves north at $4\,\mathrm{m/s}$ relative to the water, while the water flows east at $3\,\mathrm{m/s}$. What is the direction of the boat's velocity relative to the ground?

Accepted Answer

$	heta = 	an^{-1} \left( \dfrac{3}{4} ight)$ \, east of north

Question 15

A projectile is launched at $v_0$ and angle $	heta$. What is the vertical velocity $v_y$ at time $t$?

Accepted Answer

$v_y = v_0 \sin 	heta - g t$

Question 16

A ball is thrown horizontally from a $45\,\mathrm{m}$ high cliff. How long does it take to reach the ground? (Use $g = 9.8\,\mathrm{m/s^2}$)

Accepted Answer

$t = \sqrt{\dfrac{2 	imes 45}{9.8}} \approx 3.03\,\mathrm{s}$

Question 17

A projectile is launched at $v_0$ and angle $	heta$. What is the formula for its horizontal displacement $\Delta x$ after time $t$?

Accepted Answer

$\Delta x = v_0 \cos 	heta \cdot t$

Question 18

A ball is thrown at $v_0$ and angle $	heta$. At what point in its trajectory is its speed minimum?

Accepted Answer

At the maximum height

Question 19

A projectile is launched at $v_0$ and angle $	heta$. What is the formula for the time to reach maximum height?

Accepted Answer

$t_{max} = \dfrac{v_0 \sin 	heta}{g}$

Physics I: 2D Kinematics and Relative Velocity – Study Notes

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