Q: A projectile is launched with an initial velocity $$v_0 at $$angle $$\theta$$ above the horizontal. What is the time of flight?

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

Q: A projectile is launched at $$v_0$$ and angle $$\theta. $$What is the equation for its vertical displacement $$y as a $$function of time $$t$$?

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

Question 1

A soccer ball is kicked from the ground with an initial velocity of $$20\ 	ext{m/s} at an $$angle of $$30^\circ$$ above the horizontal. Neglecting air resistance, what is the maximum height reached by the ball?

Accepted Answer

$$5.1\ 	ext{m}$$

Question 2

A projectile is launched from the top of a $$50\ 	ext{m}$$ cliff with a velocity of $$15\ 	ext{m/s}$$ horizontally. How long does it take to hit the ground?

Accepted Answer

$$3.2\ 	ext{s}$$

Question 3

Which of the following best describes the acceleration of a projectile in two dimensions (neglecting air resistance)?

Accepted Answer

Constant and directed downward with magnitude $$g$$

Question 4

A car moves along a circular track of radius $$100\ 	ext{m} at a $$constant speed of $$20\ 	ext{m/s}. $$What is the magnitude of its acceleration?

Accepted Answer

$$4\ 	ext{m/s}^2$$

Question 5

A stone is thrown at an angle of $$45^\circ$$ with an initial speed $$v_0. $$What is the horizontal range $$R of $$the stone (neglecting air resistance)?

Accepted Answer

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

Question 6

A drone flies with a velocity of $$\vec{v} = 5\hat{i} + 3\hat{j}\ 	ext{m/s}. $$What is the magnitude of its velocity?

Accepted Answer

$$\sqrt{34}\ 	ext{m/s}$$

Question 7

A ball is thrown horizontally from a $$40\ 	ext{m}$$ high building with a speed of $$10\ 	ext{m/s}. $$How far from the base of the building does the ball land?

Accepted Answer

$$28.4\ 	ext{m}$$

Question 8

Which equation correctly gives the position vector $$\vec{r}(t)$$ for a particle moving in two dimensions with constant acceleration $$\vec{a}$$?

Accepted Answer

$$\vec{r}(t) = \vec{r}_0 + \vec{v}_0 t + \dfrac{1}{2} \vec{a} t^2$

Question 9

A projectile is launched with an initial velocity $$v_0 at $$angle $$	heta$$ above the horizontal. What is the time of flight?

Accepted Answer

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

Question 10

A cyclist moves in a straight line with velocity $$\vec{v}(t) = (2t)\hat{i} + (3)\hat{j}$$, where $$t is in $$seconds. What is the acceleration vector?

Accepted Answer

$$\vec{a} = 2\hat{i}$$

Question 11

A plane drops a package while flying horizontally at $$100\ 	ext{m/s} at a $$height of $$500\ 	ext{m}. $$How far horizontally from the drop point does the package land?

Accepted Answer

$$1000\ 	ext{m}$$

Question 12

A particle moves in three dimensions with position $$\vec{r}(t) = (2t)\hat{i} + (3t^2$)\hat{j} + (4)\hat{k}. $$What is its velocity vector at $$t = 1\ 	ext{s}$$?

Accepted Answer

$$\vec{v} = 2\hat{i} + 6\hat{j}$$

Question 13

A projectile is launched at $$v_0 = 25$$\ 	ext{m/s}$$ at 60$$^\circ$$ above the horizontal. What is the vertical component of its initial velocity?

Accepted Answer

$$21.7\ 	ext{m/s}$$

Question 14

A runner moves along a path described by $$\vec{r}(t) = (4t)\hat{i} + (t^2$)\hat{j}. $$What is the speed at $$t = 2\ 	ext{s}$$?

Accepted Answer

$$\sqrt{32}\ 	ext{m/s}$$

Question 15

A projectile is launched from ground level with initial velocity $$v_0 at $$angle $$	heta. $$Which statement is true about the horizontal velocity during flight (neglecting air resistance)?

Accepted Answer

It remains constant throughout the flight.

Question 16

A ball is thrown with velocity $$\vec{v}_0 = 10\hat{i} + 20\hat{j}\ 	ext{m/s}. $$What is the initial speed?

Accepted Answer

$$\sqrt{500}\ 	ext{m/s}$$

Question 17

A particle moves in a circle of radius $$r at $$constant speed $$v. $$What is the direction of its acceleration?

Accepted Answer

Toward the center of the circle

Question 18

A projectile is launched at $$v_0$$ and angle $$	heta. $$What is the equation for its vertical displacement $$y as a $$function of time $$t$$?

Accepted Answer

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

Question 19

A rescue helicopter drops a supply package from a height of $$80\ 	ext{m}$$ while moving horizontally at $$30\ 	ext{m/s}. $$How long does it take for the package to reach the ground?

Accepted Answer

$$4.02\ 	ext{s}$$

Kinematics in Two and Three Dimensions: Study Notes

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