One-Dimensional Motion in the Vertical Direction (y – axis) or Freely Falling Bodies Montwood High School Physics R. Casao

Freely Falling Bodies Any object that is released into the air is subject to a vertical acceleration due to the attractive gravitational force between the object and the Earth. The direction of the acceleration is downward toward the surface of the Earth. For now, we will only consider objects that are moving either straight up or straight down (along the vertical y-axis).

The vertical acceleration due to gravity has the value of 9.8 m/s 2 near the surface of the Earth. Most books and people use the letter g to indicate the acceleration due to gravity. a g = 9.8 m/s 2 The magnitude of a g decreases with increasing height above the Earth’s surface. The vector a g is directed downward toward the center of the Earth. Down is positive for a g ; up is negative for a g.

Pictorial representation of an object dropped from rest at a height of 100 m above the ground.

Pictorial representation of an object thrown upward with an initial velocity of 100 m/s.

Air Resistance Air resistance is the flow of air around an object and it acts to slow the velocity of a moving object. We will ignore the effect of air resistance on the objects. In the absence of air resistance, all objects dropped near the surface of the Earth change velocity with a constant acceleration. –All objects travel the same distance in the same amount of time (which is also true in the presence of air resistance as long as the air resistance on both objects is the same). All objects released from the same height at the same time will strike the ground at the same time. –This motion is called free fall.

A freely falling object is any object moving freely under the influence of gravity. Objects thrown upward or downward and those released from rest (dropped) are all freely falling objects once they are released. An object thrown upward or downward will experience the same acceleration as an object released from rest. Once they are in free fall, all objects will have an acceleration downward of -9.8 m/s 2.

When water is inside a container and the container has a hole in the side, the water will escape out the side of the hole. Gravity pulls downward on both the container and the water in the container. The hand pulls upward on the container to keep it from falling. The bottom of the container pushes upward on the water directly over it. The sides of the container help to keep the water in the container against the pull of gravity. At the location of the hole, the container cannot keep the water in and gravity pulls it out. When the container is dropped, both the container and the water in it are in free fall and are accelerating downward. The pull of gravity is the same on both the container and the water and the water does not come out of the container.

When air is in the tube, air resistance acts on both the rock and the paper. Air resistance has a greater effect on the paper, therefore, the rock strikes the bottom first. When air is removed from the tube, there is no air resistance to act on either the rock or the paper and both the rock and the paper strike the bottom at the same time.

Free-falling objects are in a state of acceleration. They are accelerating at a rate of 9.8 m/s 2. This means that the velocity of a free- falling object is changing by 9.8 m/s every second. If dropped from a position of rest, the object will be traveling 9.8 m/s at the end of the first second, 19.6 m/s at the end of the second second, 29.4 m/s at the end of the third second, etc.

This is a flash photograph of two balls released at the same time. Each image of the balls represents an equal time period of 1/30th of a second. Both balls are falling freely, but one ball (the ball on the right) was projected horizontally. Notice that the horizontal displacement of this ball does not change in equal periods of time. The ball moves an equal distance to the right for each successive time period.

In the horizontal direction there is no external force and therefore no acceleration. We can see that the horizontal velocity of the projected ball is constant. In the vertical direction, there is a noticeable acceleration. We can see this because for each successive time period, the vertical displacement of each ball increases. The change in vertical displacement is the same for both. Therefore, the vertical velocities for each will be equal.

The pull of gravity on an object is what causes an object thrown up into the air to slow down to 0 m/s (at its highest point) and then begin to increase its speed as it falls to Earth. Even when the velocity is zero, the acceleration is –9.8 m/s 2. Only the magnitude and direction of the velocity changes.

For an object which is released from one point, rises upward to a maximum height, then falls downward to return to its initial position: Time up = time down Total time = time up + time down v yi = -v yf The velocity at the highest point is 0 m/s. The acceleration at every point, including the highest point, is –9.8 m/s 2.

For an object thrown into the air and under the acceleration of gravity, the velocity at a point on the way up and the velocity at the same point on the way down are equal in magnitude, but opposite in direction. Upward velocities are positive Downward velocities are negative

Equations If we neglect air resistance and assume that the gravitational acceleration does not change much with altitude, the motion of a freely falling body can be described as motion in one-dimension with constant acceleration. The vertical direction will be the y-axis with up being positive and down being negative. Using the equations for acceleration along the x-axis and replacing x with y and a with – a g :

The negative signs in these equations indicate the downward direction of the gravitational acceleration.

If the direction of motion is upward, then  y is positive; if the direction of motion is downward, then  y is negative. For an object thrown upward: – Final velocity is 0 m/s (at the highest point). – Acceleration: a g = 9.8 m/s 2 downward. – Initial velocity is greater than 0 m/s. – Maximum height will be equal to  y. For an object that is falling downward: – Initial velocity is 0 m/s if the object is dropped. – Initial velocity is greater than 0 m/s and is negative if the object is thrown downward.

– Acceleration: a g = 9.8 m/s 2 downward. – Distance object falls is negative and is equal to  y. – The final velocity when the object first contacts the ground will be negative and will be greater than 0 m/s. V y f is the velocity the object has as it first makes contact with the surface and it is NOT 0 m/s. The object will eventually slow down from v y f to 0 m/s as it interacts with the surface, but the object first strikes the surface with velocity v y f. Velocities in the upward direction are positive. Velocities in the downward direction are negative.

For an object that is falling downward, the  y is negative. When using the v y f 2 = v y i 2 – (2·a g ·  y) equation, the final velocity will be positive because you cannot take the square root of a negative number. You will have to add the negative sign to show that the object is moving downward. For an object that is thrown upward from a point and then lands at that point:  v y i = - v y f  time up = time down  Total time = time up + time down  Distance up = distance down

– All of this is true because the object slows down (deceleration) on the way up at the same rate as it speeds up on the way down. Both the deceleration and the acceleration are constant and due to the force of gravity pulling on the object. For Problems Involving Ascending (Rising) or Descending (Moving Downward) Objects: – These types of problems have an item being released or falling off of an object (I will use a hot air balloon as an example) that is either rising or coming down with some speed. The key to solving these problems is to realize that the balloon and any object in the balloon or attached to the balloon is traveling at the same speed at the balloon.

For objects that are rising: – If a balloon is rising at 10 m/s and an object is released from the balloon. The released object will have an upward velocity of 10 m/s and will be decelerated by gravity as it travels upward. The final velocity on the way up is 0 m/s. The upward distance  y traveled by the object will be positive. – Both the upward distance  y and the time up can be determined from this information.

– After the object reaches its highest point, gravity will begin to accelerate the object down toward the ground. The initial velocity is 0 m/s and the distance the object falls will be the distance above the ground at which the object left the balloon plus the upward distance  y traveled by the object to the highest point. The  y down will be negative to indicate that the motion is downward. –

From this information, you can determine the time down and the velocity at which the object will strike the surface (v y f ). The final velocity should be negative to indicate that the direction of motion is downward. The total time in the air will be equal to the time up + the time down. The maximum height will be equal to the height of release +  y up.

For objects that are hovering (staying on one place; not rising or moving downward): – The initial velocity will be zero and you can solve this problem as you would for an object that is dropped downward. The  y down will be negative to indicate that the motion is downward. – From this information, you can determine the time down and the velocity at which the object will strike the surface (v f ). The final velocity should be negative to indicate that the direction of motion is downward.

For objects that are descending (moving downwards) when the object is released: If a balloon is descending at 10 m/s and an object is released from the balloon. The released object will have a downward velocity of -10 m/s and will be accelerated by gravity as it travels downward. The initial velocity v i and the  y down will be negative to indicate that the motion is downward. From this information, you can determine the time down and the velocity at which the object will strike the surface (v y f ). The final velocity should be negative to indicate that the direction of motion is downward.

Graphs of Free Fall Motion Position – time graph for an object dropped from a position above the floor. Velocity – time graph for an object dropped from a position above the floor.

Graphs of Free Fall Motion Position – time graph for an object thrown up and returning to the position from which it was thrown.

Velocity Graphs for Vertical Upward & Downward Motion in Free-Fall t Vy t Vx Vertical motion begins in the positive direction at a maximum and decreases as the object rises; the velocity is 0 m/s at the highest point, and then increases in the negative direction as the object falls. The horizontal velocity is 0 m/s.

Acceleration Graphs for Free-Fall t ay t ax Vertical acceleration a y is – 9.8 m/s 2. Horizontal acceleration a x is 0 m/s 2.