c. projectile range as a function of angle\nwe will investigate projectile range as a function of launch…

c. projectile range as a function of angle\nwe will investigate projectile range as a function of launch angle. we will fire a projectile at a range of angles and analyze its motion. below, derive the expression relating range ($x_{i}$) to firing angle $\theta$ and initial velocity when the projectile is launched and lands at $y = 0$. arrive at the formula: $r=\frac{v_{i}^{2}sin2\theta}{g}$.\n1. start the phet software. feel free to \play\ with the software and its various functions, but we will use the lab section displayed on the opening page. click into the lab section. on the right hand side, make sure to deselect air resistance. air resistance will remain off for this experiment. mass and diameter of the cannonball will also remain fixed. their value will not affect the outcomes we are studying. set gravity to 9.81 m/s². set initial speed to 15 m/s. for constant values of initial speed (15 m/s), fire the projectile for the angles (15, 25, 35, 45, 55, 65, 75, 85, 90 degrees). using the time, range, and height tool, collect the following data and enter into table 2. the time, range and height tool is accessible in the top right corner of the simulation and looks like this: put the cross hair on the data point of interest.\nmeasure the following quantities:\nr - range (location where projectile hits the ground)\n$t_{f}$ - flight time (time when projectile hits the ground)\nh - maximum height (peak of the projectile motion)\n$t_{h}$ - time at max height (time at peak of the projectile motion)

c. projectile range as a function of angle\nwe will investigate projectile range as a function of launch angle. we will fire a projectile at a range of angles and analyze its motion. below, derive the expression relating range ($x_{i}$) to firing angle $\theta$ and initial velocity when the projectile is launched and lands at $y = 0$. arrive at the formula: $r=\frac{v_{i}^{2}sin2\theta}{g}$.\n1. start the phet software. feel free to \play\ with the software and its various functions, but we will use the lab section displayed on the opening page. click into the lab section. on the right hand side, make sure to deselect air resistance. air resistance will remain off for this experiment. mass and diameter of the cannonball will also remain fixed. their value will not affect the outcomes we are studying. set gravity to 9.81 m/s². set initial speed to 15 m/s. for constant values of initial speed (15 m/s), fire the projectile for the angles (15, 25, 35, 45, 55, 65, 75, 85, 90 degrees). using the time, range, and height tool, collect the following data and enter into table 2. the time, range and height tool is accessible in the top right corner of the simulation and looks like this: put the cross hair on the data point of interest.\nmeasure the following quantities:\nr - range (location where projectile hits the ground)\n$t_{f}$ - flight time (time when projectile hits the ground)\nh - maximum height (peak of the projectile motion)\n$t_{h}$ - time at max height (time at peak of the projectile motion)

Answer

Explanation:

Step1: Resolve initial velocity

The initial velocity $v_i$ can be resolved into horizontal $v_{ix}=v_i\cos\theta$ and vertical $v_{iy} = v_i\sin\theta$ components.

Step2: Analyze vertical - motion

In vertical - motion, the displacement $y = v_{iy}t-\frac{1}{2}gt^{2}$. Since the projectile starts and lands at $y = 0$, we have $0=v_i\sin\theta t-\frac{1}{2}gt^{2}$. Factoring out $t$, we get $t(v_i\sin\theta-\frac{1}{2}gt)=0$. One solution is $t = 0$ (corresponds to the launch time), and the other is $t=\frac{2v_i\sin\theta}{g}$ (corresponds to the time of landing).

Step3: Analyze horizontal - motion

In horizontal - motion (with constant velocity since there is no acceleration in the horizontal direction in the absence of air - resistance), the range $R=v_{ix}t$. Substituting $v_{ix}=v_i\cos\theta$ and $t = \frac{2v_i\sin\theta}{g}$ into the range formula, we get $R=v_i\cos\theta\times\frac{2v_i\sin\theta}{g}$.

Step4: Use double - angle formula

Using the double - angle formula $\sin2\theta = 2\sin\theta\cos\theta$, we can rewrite the range formula as $R=\frac{v_i^{2}\sin2\theta}{g}$.

Answer:

The derivation of the range formula $R=\frac{v_i^{2}\sin2\theta}{g}$ is shown above.