Q: A projectile is launched with an initial speed $$v_0 at an $$angle $$\theta$$ above the horizontal. What is the time of flight until it returns to the same vertical level?

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

Question 1

A rocket car starts at $$x = 0$$ with an initial velocity of $$3.00\ \mathrm{m/s}$$ and accelerates according to $$a(t) = 9.00\ \mathrm{m/s^3$} \ t. $$What is the velocity of the car at $$t = 2.00\ \mathrm{s}$$?

Accepted Answer

$$v = 3.00\ \mathrm{m/s} + \$$int_0$$^{2.00} 9.00\ t\ dt = 3.00 + 9.00\cdot1.00 = 12.0\ \mathrm{m/s}$$

Question 2

A rocket car starts at $$x = 0$$ with an initial velocity of $$3.00\ \mathrm{m/s}$$ and accelerates according to $$a(t) = 9.00\ \mathrm{m/s^3$} \ t. $$What is the position of the car at $$t = 2.00\ \mathrm{s}$$?

Accepted Answer

$$x = 0 + \$$int_0$$^{2.00} [3.00 + 4.50\ t^2$] dt = 0 + 3.00\cdot2.00 + 4.50\cdot\frac{8}{3} = 6.00 + 12.0 = 18.0\ \mathrm{m}$$

Question 3

A projectile is launched horizontally from a cliff with an initial speed of $$10.0\ \mathrm{m/s}$$ and lands $$45.2\ \mathrm{m}$$ from the base after $$4.52\ \mathrm{s}. $$From what height was it launched? (Assume $$g = 9.80\ \mathrm{m/s^2$})$$

Accepted Answer

$$y = \frac{1}{2}gt^2$ = \frac{1}{2}\cdot9.80\$$cdot(4.52)^2$$ = 100\ \mathrm{m}$$

Question 4

A basketball player moves with the velocity shown in the figure (velocity vs. time graph with segments of positive, then negative, then zero velocity). What does the area under the velocity-time graph represent?

Accepted Answer

The displacement of the player over the time interval

Question 5

Given the parametric equations $$x = 2.0 - 0.25 t^2$ and y = 1.0 t + 0.025$$ $$t^3 ($$with SI units), what is the velocity vector at $$t = 2.0\ \mathrm{s}$$?

Accepted Answer

$$\vec{v} = \left(\frac{dx}{dt}, \frac{dy}{dt}\right) = (-1.0, 1.3)\ \mathrm{m/s}$$

Question 6

A projectile is fired at an angle to the horizontal. Which of the following statements is true about its motion (neglecting air resistance)?

Accepted Answer

The horizontal and vertical motions are independent of each other.

Question 7

A ball is dropped from rest and another is shot horizontally from the same height at the same instant. Which hits the ground first?

Accepted Answer

Both hit the ground at the same time.

Question 8

A rover on Mars has its position given by $$x = 2.0 - 0.25 t^2$ and y = 1.0 t + 0.025$$ $$t^3. $$What is the acceleration vector at $$t = 1.0\ \mathrm{s}$$?

Accepted Answer

$$\vec{a} = \left(\frac{d^2$x}{dt^2$}, \frac{d^2$y}{dt^2$}\right) = (-0.5, 0.15)\ \mathrm{m/s^2$}$$

Question 9

A projectile is launched with an initial speed $$v_0 at an $$angle $$	heta$$ above the horizontal. What is the time of flight until it returns to the same vertical level?

Accepted Answer

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

Question 10

A car moves with velocity $$v(t) = 3.00 + 4.50 t^2$ (in m/s). What is the displacement between t = 0$ and t = 2.00$$\ \mathrm{s}$$?

Accepted Answer

$$\Delta x = \$$int_0$$^{2.00} (3.00 + 4.50 t^2$) dt = 3.00\cdot2.00 + 4.50\cdot\frac{8}{3} = 6.00 + 12.0 = 18.0\ \mathrm{m}$$

Question 11

A projectile is launched horizontally from a height $$h$$ and lands a distance $$d$$ from the base after time $$t. $$Which equation correctly relates $$d$$, $$v_0$$, and $$t$$?

Accepted Answer

$$d = v_0$ t$$

Question 12

A car accelerates from rest with constant acceleration $$a. $$Which equation gives the distance traveled after time $$t$$?

Accepted Answer

$$x = \frac{1}{2} a t^2$

Question 13

A projectile is launched at $$18\ \mathrm{m/s} at an $$angle of $$80^\circ$$ above the horizontal. Which component of velocity is responsible for the range of the projectile?

Accepted Answer

The horizontal component, $$v_0$$ \cos	heta$$

Question 14

A student sketches the path of a rover on Mars given by $$x = 2.0 - 0.25 t^2$ and y = 1.0 t + 0.025$$ $$t^3. $$What is the initial direction of motion at $$t = 0$$?

Accepted Answer

Upward and to the right

Question 15

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

Accepted Answer

Constant and directed downward

Calculus-Based Motion: 1D and 2D Kinematics & Projectile Motion

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