# Problem: To understand how the trajectory of an object depends on its initial velocity, and to understand how air resistance affects the trajectory. For this problem, use the PhET simulation Projectile Motion. This simulation allows you to fire an object from a cannon, see its trajectory, and measure its range and hang time (the amount of time in the air). Click to launch video simulation Start the simulation. Press Fire to launch an object. You can choose the object by clicking on one of the objects in the scroll-down menu at top right (a cannonball is not among the choices). To adjust the cannon barrel’s angle, click and drag on it or type in a numerical value (in degrees). You can also adjust the speed, mass, and diameter of the object by typing in values. Clicking Air Resistance displays settings for (1) the drag coefficient and (2) the altitude (which controls the air density). For this tutorial, we will use an altitude of zero (sea level) and let the drag coefficient be automatically set when the object is chosen. Play around with the simulation. When you are done, click Erase and select a baseball prior to beginning Part A. Leave Air Resistance unchecked. In the previous part, you found that a ball fired with an initial speed of 25 m/s and an angle of 53° reaches the same height as a ball fired vertically with an initial speed of 20 m/s. Which ball takes longer to land?

###### FREE Expert Solution

Let's set up the position equation of the ball using UAM equation number 3 — assume the origin of our coordinate system is located at the launch point.

For the first case:

• y0 = 0 m
• yf = -1.2 m
• g = 9.8 m
• v0,y = 20 m/s

yf = y0 + v0,yt1 - (1/2)gt12

-1.2 = 0 + 20t1 - 4.9t12

0 = 1.2 + 20t1 - 4.9t12

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###### Problem Details

To understand how the trajectory of an object depends on its initial velocity, and to understand how air resistance affects the trajectory. For this problem, use the PhET simulation Projectile Motion. This simulation allows you to fire an object from a cannon, see its trajectory, and measure its range and hang time (the amount of time in the air). Click to launch video simulation Start the simulation. Press Fire to launch an object. You can choose the object by clicking on one of the objects in the scroll-down menu at top right (a cannonball is not among the choices). To adjust the cannon barrel’s angle, click and drag on it or type in a numerical value (in degrees). You can also adjust the speed, mass, and diameter of the object by typing in values. Clicking Air Resistance displays settings for (1) the drag coefficient and (2) the altitude (which controls the air density). For this tutorial, we will use an altitude of zero (sea level) and let the drag coefficient be automatically set when the object is chosen. Play around with the simulation. When you are done, click Erase and select a baseball prior to beginning Part A. Leave Air Resistance unchecked.

In the previous part, you found that a ball fired with an initial speed of 25 m/s and an angle of 53° reaches the same height as a ball fired vertically with an initial speed of 20 m/s. Which ball takes longer to land?