Q: In the experiment, why is it important to ensure that the height from which the projectile is launched and the height at which it lands are equal when using the range formula?

So that the range formula $$R = \frac{v_i^2$ \sin 2\theta}{g} is $$valid

Q: If a projectile is launched at an angle $$\theta$$ and lands at the same height from which it was launched, what is the correct formula for its range $$R$$?

$$R = \frac{v_i^2$ \sin 2\theta}{g}$$

Q: If you are given $$v_{x_i}$$ and $$\theta$$, which equation allows you to solve for the initial speed $$v_i$$?

$$v_i = \frac{v_{x_i}}{\cos \theta}$$

Question 1

A projectile is launched horizontally from a table and lands a horizontal distance $$x_f$$ from the launch point after a time $$\Delta t. $$Which equation correctly relates these quantities, assuming no air resistance?

Accepted Answer

$$x_f = v_{x_i} \Delta t$$

Question 2

If a projectile is launched at an angle $$	heta$$ with initial speed $$v_i$$, what is the correct expression for the initial horizontal velocity component $$v_{x_i}$$?

Accepted Answer

$$v_{x_i} = v_i \cos 	heta$$

Question 3

Given the equation $$y_f = y_i + (v_{y_i}) \Delta t - \frac{1}{2}g(\Delta t)^2$, what does the term $$-\frac{1}{2}g(\Delta $$t)^2$$ represent?

Accepted Answer

The vertical displacement due to gravity during time $$\Delta t$$

Question 4

A student measures the horizontal distance $$x_f$$ and the time of flight $$\Delta t$$ for a projectile launched horizontally. Which formula should they use to calculate the initial horizontal velocity $$v_{x_i}$$?

Accepted Answer

$$v_{x_i} = \frac{x_f}{\Delta t}$$

Question 5

In the experiment, why is it important to ensure that the height from which the projectile is launched and the height at which it lands are equal when using the range formula?

Accepted Answer

So that the range formula $$R = \frac{v_i^2$ \sin 2	heta}{g} is $$valid

Question 6

If a projectile is launched at an angle $$	heta$$ and lands at the same height from which it was launched, what is the correct formula for its range $$R$$?

Accepted Answer

$$R = \frac{v_i^2$ \sin 2	heta}{g}$$

Question 7

A projectile is launched at $$30^\circ$$ above the horizontal with an initial speed of $$10\ \mathrm{m/s}. $$What is the initial vertical velocity component $$v_{y_i}$$?

Accepted Answer

$$v_{y_i} = 10 \sin 30^\circ = 5\ \mathrm{m/s}$$

Question 8

In the lab, you are asked to derive the equation $$v_{y_i} = \frac{\frac{1}{2}g(\Delta t)^2$ - y_f}{\Delta t}. $$Which of the following is a correct first step in this derivation, starting from $$y_f = y_i + (v_{y_i}) \Delta t - \frac{1}{2}g(\Delta t)^2$ and assuming y_i = 0$?

Accepted Answer

Rearrange to solve for $$v_{y_i}$$: $$v_{y_i} = \frac{\frac{1}{2}g(\Delta t)^2$ - y_f}{\Delta t}$$

Question 9

Suppose you measure $$x_f = 2.0\ \mathrm{m}$$ and $$\Delta t = 0.5\ \mathrm{s}$$ for a horizontally launched projectile. What is the initial horizontal velocity $$v_{x_i}$$?

Accepted Answer

$$v_{x_i} = \frac{2.0}{0.5} = 4.0\ \mathrm{m/s}$$

Question 10

In the experiment, why is it necessary to use at least three different launch angles separated by at least $$20^\circ$$?

Accepted Answer

To ensure a range of data for more accurate calculation of $$v_i$$

Question 11

If you are given $$v_{x_i}$$ and $$	heta$$, which equation allows you to solve for the initial speed $$v_i$$?

Accepted Answer

$$v_i = \frac{v_{x_i}}{\cos 	heta}$$

Question 12

A projectile is launched from a height $$y_i$$ and lands at $$y_f = 0. If$$ $$\Delta t is $$the time of flight, which equation gives the initial vertical velocity $$v_{y_i}$$?

Accepted Answer

$$v_{y_i} = \frac{\frac{1}{2}g(\Delta t)^2$ - y_f}{\Delta t}$$

Question 13

If you want to hit a target at a known horizontal distance $$R$$ using a projectile launcher, which equation should you use to calculate the required launch angle $$	heta$$?

Accepted Answer

$$	heta = \frac{1}{2} \sin$$^{-1}$$ \left( \frac{gR}{v_i^2$} ight)$$

Question 14

A projectile is launched at $$45^\circ$$ with an initial speed $$v_i. $$What is the value of $$\sin 2	heta in $$the range formula?

Accepted Answer

$$\sin 2	heta = \sin 90^\circ = 1$$

Question 15

If the initial speed of a projectile is doubled, how does the range $$R$$ change (assuming the same launch angle and $$y_i = y_f$$)?

Accepted Answer

The range increases by a factor of 4

Question 16

In a real-world scenario, which factor would most likely cause the actual range of a projectile to be less than the calculated value using $$R = \frac{v_i^2$ \sin 2	heta}{g}$$?

Accepted Answer

Air resistance

Question 17

A student launches a projectile and measures $$v_{x_i} = 3.0\ \mathrm{m/s}$$ and $$	heta = 60^\circ. $$What is the initial speed $$v_i$$?

Accepted Answer

$$v_i = \frac{3.0}{\cos 60^\circ} = 6.0\ \mathrm{m/s}$$

Question 18

Which of the following best explains why the time of flight for a projectile depends only on the vertical motion and not on the horizontal velocity?

Accepted Answer

Gravity acts only in the vertical direction, so vertical and horizontal motions are independent

Projectile Motion Lab: Study Notes and Analysis

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