Q: If a projectile is launched at an angle $$\theta$$ with initial velocity $$v_{i}$$, which equation gives the range $$R of $$the projectile (assuming $$y_{i} = 0$$)?

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

Q: A projectile is launched at an angle $$\theta$$ and lands a horizontal distance $$R$$ away. If you know $$R$$ and $$v_{i}$$, which equation allows you to solve for the angle $$\theta$$ required to hit a target at distance $$R$$?

$$\theta = \frac{1}{2} \sin$$^{-1}$$ \left( \frac{gR}{v_{i}^{2}} \right)$$

Question 1

A projectile is launched horizontally from a table. If the initial velocity is $$v_{i}$$ and the time of flight is $$\Delta t$$, which equation correctly gives the horizontal distance $$x_{f}$$ traveled by the projectile?

Accepted Answer

$$x_{f} = v_{i} \cdot \Delta t$$

Question 2

In the experiment, why is it important to ensure the ball rolls out of the launcher at the same height each time?

Accepted Answer

To keep the time of flight constant for each trial

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

Question 4

If a projectile is launched at an angle $$	heta$$ with initial velocity $$v_{i}$$, which equation gives the range $$R of $$the projectile (assuming $$y_{i} = 0$$)?

Accepted Answer

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

Question 5

A student measures the horizontal distance $$x_{f}$$ and the time of flight $$\Delta t$$ for a projectile. Which equation should they use to calculate the initial horizontal velocity $$v_{x}$$?

Accepted Answer

$$v_{x} = \frac{x_{f}}{\Delta t}$$

Question 6

When launching a projectile at an angle, which trigonometric function relates the initial velocity $$v_{i} to $$its vertical component $$v_{y}$$?

Accepted Answer

$$v_{y} = v_{i} \sin 	heta$$

Question 7

If the time of flight for a projectile is $$\Delta t$$ and the initial vertical velocity is zero, which equation gives the acceleration due to gravity $$g in $$terms of the vertical displacement $$y_{i}$$?

Accepted Answer

$$g = \frac{2y_{i}}{(\Delta t$$)^{2}$$}$$

Question 8

A projectile is launched at an angle $$	heta$$ and lands a horizontal distance $$R$$ away. If you know $$R$$ and $$v_{i}$$, which equation allows you to solve for the angle $$	heta$$ required to hit a target at distance $$R$$?

Accepted Answer

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

Question 9

In the experiment, why is it necessary to choose three launch angles that differ by at least $$20^{\circ}$$?

Accepted Answer

To ensure a wide range of data for analyzing projectile motion

Question 10

If you measure the vertical drop $$y_{i}$$ and the time of flight $$\Delta t$$ for a projectile, which equation allows you to solve for the initial vertical velocity $$(v_{y})_{i}$$?

Accepted Answer

$$(v_{y})_{i} = \frac{y_{i} + \frac{1}{2}g(\Delta t$$)^{2}$$}{\Delta t}$$

Question 11

A student calculates the average of three measured initial velocities $$v_{i}$$ from different trials. What is the main purpose of this calculation?

Accepted Answer

To obtain a more accurate estimate of the true initial velocity

Question 12

If the horizontal distance from the launcher to the target is $$R$$, and the initial velocity is $$v_{i}$$, which equation should be used to calculate the angle $$	heta to $$hit the target?

Accepted Answer

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

Question 13

Why is it important to measure the horizontal distance from the launching end of the projectile launcher to the center of the target when aiming?

Accepted Answer

To ensure the calculated angle $$	heta$$ will result in the projectile hitting the target

Question 14

If a projectile is launched at an angle $$	heta$$ and lands at a distance $$R$$, which variable must be measured to verify the accuracy of the range equation $$R = \frac{v_{i}^{2} \sin 2	heta}{g}$$?

Accepted Answer

The initial velocity $$v_{i}$$

Question 15

A student uses the equation $$\cos 	heta = \frac{v_{x}}{v_{i}} to $$solve for $$v_{i}. $$What is the correct rearrangement of this equation?

Accepted Answer

$$v_{i} = \frac{v_{x}}{\cos 	heta}$$

Question 16

In the experiment, why is carbon paper used under the blank paper where the projectile lands?

Accepted Answer

To mark the exact spot where the ball lands for accurate measurement

Question 17

If the initial vertical velocity $$(v_{y})_{i} is $$zero, which equation gives the time of flight $$\Delta t$$ for a projectile dropped from height $$y_{i}$$?

Accepted Answer

$$\Delta t = \sqrt{\frac{2y_{i}}{g}}$$

Question 18

A student wants to hit a target at a known distance $$R$$ using a projectile launcher. Which two measurements are essential for calculating the required launch angle?

Accepted Answer

The initial velocity $$v_{i}$$ and the horizontal distance $$R$$

Projectile Motion Lab: Analysis and Calculations

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