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One-dimensional motion involving gravity

The best way to see the basic features of motion involving gravity is to start with the simplest situations and then progress toward more complex ones. So we start by considering straight up and down motion with no air resistance or friction. These assumptions mean that the velocity (if there is any) is vertical. If the object is dropped, we know the initial velocity is zero. Once the object has left contact with whatever held or threw it, the object is in free-fall. Under these circumstances, the motion is one-dimensional and has constant acceleration of magnitude g size 12{g} {} . We will also represent vertical displacement with the symbol y size 12{y} {} and use x size 12{x} {} for horizontal displacement.

Kinematic equations for objects in free-fall where acceleration = - g

v = v 0 - gt size 12{v=v rSub { size 8{0} } + ital "gt"} {}
y = y 0 + v 0 t - 1 2 gt 2 size 12{y=y rSub { size 8{0} } +v rSub { size 8{0} } t+ { {1} over {2} } ital "gt" rSup { size 8{2} } } {}
v 2 = v 0 2 - 2 g y y 0 size 12{v rSup { size 8{2} } =v rSub { size 8{0} } rSup { size 8{2} } +2g left (y - y rSub { size 8{0} } right )} {}

Calculating position and velocity of a falling object: a rock thrown upward

A person standing on the edge of a high cliff throws a rock straight up with an initial velocity of 13.0 m/s . The rock misses the edge of the cliff as it falls back to earth. Calculate the position and velocity of the rock 1.00 s, 2.00 s, and 3.00 s after it is thrown, neglecting the effects of air resistance.

Strategy

Draw a sketch.

Velocity vector arrow pointing up in the positive y direction, labeled v sub 0 equals thirteen point 0 meters per second. Acceleration vector arrow pointing down in the negative y direction, labeled a equals negative 9 point 8 meters per second squared.

We are asked to determine the position y size 12{y} {} at various times. It is reasonable to take the initial position y 0 size 12{y rSub { size 8{0} } } {} to be zero. This problem involves one-dimensional motion in the vertical direction. We use plus and minus signs to indicate direction, with up being positive and down negative. Since up is positive, and the rock is thrown upward, the initial velocity must be positive too. The acceleration due to gravity is downward, so a size 12{a} {} is negative. It is crucial that the initial velocity and the acceleration due to gravity have opposite signs. Opposite signs indicate that the acceleration due to gravity opposes the initial motion and will slow and eventually reverse it.

Since we are asked for values of position and velocity at three times, we will refer to these as y 1 size 12{y rSub { size 8{1} } } {} and v 1 size 12{v rSub { size 8{1} } } {} ; y 2 size 12{y rSub { size 8{2} } } {} and v 2 size 12{v rSub { size 8{2} } } {} ; and y 3 size 12{y rSub { size 8{3} } } {} and v 3 size 12{v rSub { size 8{3} } } {} .

Solution for Position y 1 size 12{y rSub { size 8{1} } } {}

1. Identify the knowns. We know that y 0 = 0 size 12{y rSub { size 8{0} } =0} {} ; v 0 = 13 . 0 m/s size 12{v rSub { size 8{0} } ="13" "." "0 m/s"} {} ; a = g = 9 . 80 m/s 2 size 12{a= - g= - 9 "." "80 m/s" rSup { size 8{2} } } {} ; and t = 1 . 00 s size 12{t=1 "." "00 s"} {} .

2. Identify the best equation to use. We will use y = y 0 + v 0 t + 1 2 at 2 size 12{y=y rSub { size 8{0} } +v rSub { size 8{0} } t+ { {1} over {2} } ital "at" rSup { size 8{2} } } {} because it includes only one unknown, y size 12{y} {} (or y 1 size 12{y rSub { size 8{1} } } {} , here), which is the value we want to find.

3. Plug in the known values and solve for y 1 size 12{y rSub { size 8{1} } } {} .

y 1 = 0 + 13 . 0 m/s 1 . 00 s + 1 2 9 . 80 m/s 2 1 . 00 s 2 = 8 . 10 m size 12{y"" lSub { size 8{1} } =0+ left ("13" "." "0 m/s" right ) left (1 "." "00 s" right )+ { {1} over {2} } left ( - 9 "." "80"" m/s" rSup { size 8{2} } right ) left (1 "." "00 s" right ) rSup { size 8{2} } =8 "." "10"`m} {}

Discussion

The rock is 8.10 m above its starting point at t = 1 . 00 size 12{t=1 "." "00"} {} s, since y 1 > y 0 size 12{y rSub { size 8{1} }>y rSub { size 8{0} } } {} . It could be moving up or down; the only way to tell is to calculate v 1 size 12{v rSub { size 8{1} } } {} and find out if it is positive or negative.

Solution for Velocity v 1 size 12{v rSub { size 8{1} } } {}

1. Identify the knowns. We know that y 0 = 0 size 12{y rSub { size 8{0} } =0} {} ; v 0 = 13 . 0 m/s size 12{v rSub { size 8{0} } ="13" "." "0 m/s"} {} ; a = g = 9 . 80 m/s 2 size 12{a= - g= - 9 "." "80 m/s" rSup { size 8{2} } } {} ; and t = 1 . 00 s size 12{t=1 "." "00 s"} {} . We also know from the solution above that y 1 = 8 . 10 m size 12{y rSub { size 8{1} } =8 "." "10 m"} {} .

2. Identify the best equation to use. The most straightforward is v = v 0 gt size 12{v=v rSub { size 8{0} } - ital "gt"} {} (from v = v 0 + at size 12{v=v rSub { size 8{0} } + ital "at"} {} , where a = gravitational acceleration = g size 12{a="gravitational acceleration"= - g} {} ).

3. Plug in the knowns and solve.

v 1 = v 0 gt = 13 . 0 m/s 9 . 80 m/s 2 1 . 00 s = 3 . 20 m/s size 12{v rSub { size 8{1} } =v rSub { size 8{0} } - ital "gt"="13" "." "0 m/s" - left (9 "." "80 m/s" rSup { size 8{2} } right ) left (1 "." "00 s" right )=3 "." "20 m/s"} {}

Discussion

The positive value for v 1 means that the rock is still heading upward at t = 1 . 00 s . However, it has slowed from its original 13.0 m/s, as expected.

Practice Key Terms 2

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Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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