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

$$v = \sqrt{(v_0$ \cos \$$theta)^2$$ + (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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