Q: Given the equation $$y_f = y_i + (v_{yi}) \triangle t - \frac{1}{2}g(\triangle t)^2$, what does the term $$-\frac{1}{2}g(\triangle $$t)^2$$ represent in the context of projectile motion?

The vertical displacement due to gravity during time $$\triangle t.$$

Q: A student measures the time $$\triangle t$$ for a projectile to hit the ground and the horizontal distance $$x_f. $$Using $$x_f = v_{xi} \triangle t$$, how should the student solve for the initial horizontal velocity $$v_{xi}$$?

$$v_{xi} = \frac{x_f}{\triangle t}$$

Q: A projectile is launched at an angle $$\theta$$ and lands a horizontal distance $$R$$ away. Which formula gives the range $$R$$ for $$y_i = y_f = 0$$?

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

Q: If you want to hit a target placed at a horizontal distance $$R$$ using your best estimate of $$v_i$$, which equation should you use to calculate the required launch angle $$\theta$$?

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

Q: If the initial vertical position $$y_i is $$not equal to the final vertical position $$y_f$$, which equation should be used to determine the initial vertical velocity $$v_{yi}$$?

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

Question 1

A projectile is launched horizontally from a table and lands on the floor a horizontal distance $$x_f$$ away after a time $$	riangle t. $$Which equation correctly relates $$x_f$$, the initial velocity $$v_i$$, and $$	riangle t$$ for this scenario?

Accepted Answer

$$x_f = v_i 	riangle t$$

Question 2

In the experiment, why is it important to ensure the ball rolls off the launcher and not out of the front slot when measuring projectile motion?

Accepted Answer

To ensure the initial velocity is consistent and the ball's trajectory is predictable.

Question 3

Given the equation $$y_f = y_i + (v_{yi}) 	riangle t - \frac{1}{2}g(	riangle t)^2$, what does the term $$-\frac{1}{2}g(	riangle $$t)^2$$ represent in the context of projectile motion?

Accepted Answer

The vertical displacement due to gravity during time $$	riangle t.$$

Question 4

If a projectile is launched at an angle $$	heta$$ with initial velocity $$v_i$$, which equation gives the horizontal component of the initial velocity?

Accepted Answer

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

Question 5

A student measures the time $$	riangle t$$ for a projectile to hit the ground and the horizontal distance $$x_f. $$Using $$x_f = v_{xi} 	riangle t$$, how should the student solve for the initial horizontal velocity $$v_{xi}$$?

Accepted Answer

$$v_{xi} = \frac{x_f}{	riangle t}$$

Question 6

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

Accepted Answer

To ensure a range of data for analyzing the effect of angle on projectile motion.

Question 7

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

Accepted Answer

The initial vertical velocity of the projectile.

Question 8

A projectile is launched at an angle $$	heta$$ and lands a horizontal distance $$R$$ away. Which formula gives the range $$R$$ for $$y_i = y_f = 0$$?

Accepted Answer

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

Question 9

If you want to hit a target placed at a horizontal distance $$R$$ using your best estimate of $$v_i$$, 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 10

During the experiment, why is it important to measure the horizontal distance from the launching end of the projectile launcher to the center of the target?

Accepted Answer

To accurately determine the range $$R$$ for calculating the required launch angle.

Question 11

If the initial vertical position $$y_i is $$not equal to the final vertical position $$y_f$$, which equation should be used to determine the initial vertical velocity $$v_{yi}$$?

Accepted Answer

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

Question 12

A student calculates the average of several initial velocity values $$v_i$$ from different trials. What is the purpose of this average in the context of the experiment?

Accepted Answer

To obtain the best estimate for the true value of $$v_i.$$

Question 13

Which trigonometric function is used to relate the initial vertical velocity $$v_{yi} to $$the initial velocity $$v_i$$ and launch angle $$	heta$$?

Accepted Answer

$$\sin 	heta = \frac{v_{yi}}{v_i}$$

Question 14

If a projectile is launched at an angle $$	heta$$ and you know $$v_i$$ and $$R$$, which equation allows you to solve for $$	heta$$?

Accepted Answer

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

Question 15

Why is it important to use carbon paper in the experiment when measuring where the projectile lands?

Accepted Answer

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

Question 16

If the projectile launcher is set at an angle $$	heta$$ and the ball lands at a distance $$R$$, which variable must be measured to ensure Equation (7) applies?

Accepted Answer

The horizontal distance from the launcher to the target ($$R).$$

Question 17

A student fires a projectile at three different angles and records the time of flight and horizontal distance for each. What is the main reason for varying the launch angle?

Accepted Answer

To analyze how the launch angle affects the range and time of flight.

Question 18

When deriving the range formula $$R = \frac{v_i^2$ \sin 2	heta}{g}$$, which assumption is made about the initial and final vertical positions?

Accepted Answer

$$y_i = y_f = 0$$

Question 19

If the ball is launched and lands at the same vertical height, which variable does NOT affect the range $$R$$?

Accepted Answer

The mass of the ball.

Projectile Motion Laboratory: Concepts, Equations, and Applications

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