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Wize University Physics Textbook (Master) > Motion in 1/2/3D: Kinematics

Free Fall

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Free Fall


  • Near the Earth's surface, all objects experience the same acceleration due to gravity (neglecting air resistance). This acceleration has magnitude g=9.81m/s2g = 9.81 m/s^2 and points downward, to the center of the Earth.

  • Note that this value changes if you are extremely far away from Earth's surface, or if you are on another planet (or moon)!

  • To solve problems with freely falling objects, we simply use g in the kinematic equations in place of the acceleration, a. If you define the downwards direction to be negative, then you will use -g. Even if the object is thrown upward, the gravitational acceleration still points downward.
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Example: Free Fall


I drop a ball off of a 100m tall building.

a) What is its velocity just before it hits the ground?
b) What is its average velocity?
c) What is the instantaneous velocity when the ball is halfway down the building?
Use g=9.81 m/s2.

Solutions:


v0=0msvf=?d=100 ma=9.81 m/s2\begin{array}{l} v_0=0\frac{m}{s}\\ v_f=?\\ d=100~m\\ a=9.81~m/s^2 \end{array}

Part a)


I choose downward direction to be positive. So, any vector pointing downward is positive and any vector pointing upward is negative. (Alternatively you can choose upward direction to be positive. As long as you remain consistent, you should get the same answer at the end)

Use the full distance, 100 meters.
vf2=v02+2advf=v02+2ad=02+2(9.81m/s2)(100m)=44.3 m/s before it hits the ground\begin{array}{lll} v_f^2=v_0^2+2ad\\ v_f=\sqrt{v_0^2+2ad}\\[3pt] =\sqrt{0^2+2(9.81 m/s^2)(100m)}\\[2pt] =44.3~m/s~\textrm{before it hits the ground} \end{array}

Part b)


When acceleration is constant, the average velocity can be determined simply by averaging the initial and final velocities:

vavg=v1 + v22=0 + 44.32=22.1 m/sv_{avg}=\frac{v_{1}\ +\ v_{2}}{2}=\frac{0\ +\ 44.3}{2}=22.1~m/s

Note: we are defining the downward direction as positive, so we leave all answers as positive. If you picked the frame of reference pointing upward, your answers would need to be answered as negative. (If instead we asked for SPEED instead of velocity, then you would enter all positive values because speed is a scalar.)



Part c)


Use half of the distance (50 meters).
vf2=v02+2adv0=0msvf=v02+2advf=?=02+2(9.81m/s2)(50m)d=100 m=31.3 m/s halfway down the buildinga=9.81 m/s2\begin{array}{lll} v_f^2=v_0^2+2ad&&v_0=0\frac{m}{s}\\ v_f=\sqrt{v_0^2+2ad}&&v_f=?\\[3pt] =\sqrt{0^2+2(9.81m/s^2)(50m)}& &d=100~m\\[3pt] =31.3~m/s~\textrm{halfway down the building} && a=9.81~m/s^2 \end{array}




A boy decided to estimate the height of a two-storey building by throwing a stone up and measuring the time it takes for the stone to reach the top of the building. Assume he throws the stone from 1 meter above the ground, such that the stone stops and turns back just when it reached the top of the building.

If the stone takes 2s to reach the top of the building, what is the initial velocity of the stone? (Pick upward direction to be positive)

Extra Practice
Kinematics: Frog Jump
A frog jumps by pushing off the ground, and accelerates very quickly while its feet are still on the ground. Its center of mass lifts 3.0 cm while it is pushing off. The frog reaches a height of 40 cm off the ground before falling again. What acceleration did it have while launching itself off the ground?
Kinematics: Frog Jump
A frog jumps by pushing off the ground, and accelerates very quickly while its feet are still on the ground. Its center of mass lifts 3.0 cm while it is pushing off. The frog reaches a height of 40 cm off the ground before falling again. What acceleration did it have while launching itself off the ground?
Kinematics: Frog Jump
A frog jumps by pushing off the ground, and accelerates very quickly while its feet are still on the ground. Its center of mass lifts 3.0 cm while it is pushing off. The frog reaches a height of 40 cm off the ground before falling again. What acceleration did it have while launching itself off the ground?
Kinematics: Frog Jump
A frog jumps by pushing off the ground, and accelerates very quickly while its feet are still on the ground. Its center of mass lifts 3.0 cm while it is pushing off. The frog reaches a height of 40 cm off the ground before falling again. What acceleration did it have while launching itself off the ground?
Kinematics: Frog Jump
A frog jumps by pushing off the ground, and accelerates very quickly while its feet are still on the ground. Its center of mass lifts 3.0 cm while it is pushing off. The frog reaches a height of 40 cm off the ground before falling again. What acceleration did it have while launching itself off the ground?
Kinematics: Frog Jump
A frog jumps by pushing off the ground, and accelerates very quickly while its feet are still on the ground. Its center of mass lifts 3.0 cm while it is pushing off. The frog reaches a height of 40 cm off the ground before falling again. What acceleration did it have while launching itself off the ground?
Kinematics: Airplane dropping supplies
An airplane is travelling at horizontal speed v, at height h. It needs to drop supplies at the right time so that residents of an island will receive them. At what distance before flying over the islanders should you drop the supplies?
(If you feel more comfortable using numbers instead of variables, first try the problem with v = 215 km/h, h = 200 m.)
A 2 kg2\ kgball is dropped from a height of 10 m10\ monto the ground.
a) If the balls experiences no losses find the velocity of the ball right before it hits the ground. (in m/s)
b) If the ball loses 25% of its energy when it bounces, what height will it reach on its first bounce? (in m)
A slingshot is used to shoot a 500g500gmetal ball straight into the air (vertically). The slingshot can store a maximum energy of 250J250Jin its elastic.
a) What is the highest a ball could be shot?
b) What is the maximum speed that the ball would reach?
Two balls of equal mass are thrown from the top of a building with height10.0m10.0 m. One is thrown straight up at 10 m/s10\ m/s(call this ball A) and the other is thrown horizontally at 10 m/s10\ m/s (call this ball B).
a) Which balls has a greater speed when it reaches the ground? Why? Make sure you include conservation of energy in your answer.
b) Your friend tries to tell you that the ball thrown horizontally would have greater speed, because it has both an x-component and a y-component, and so the speed (magnitude of the velocity) should be higher. Why is he wrong? (Calculation not required, but you can do one if you want.)
Two children, Bob and Alice, and their dog, Buttercup, jump off of a bridge into the river below. Bob weighs more than Alice. Alice weighs twice as much as Buttercup.
a) Neglecting air friction and drag, who hits the water with the highest speed?
b) What is the ratio between the amount of energy Alice has at the top of the bridge compared to Buttercup?
Practice: Velocity & Acceleration
A ball is thrown with an initial upwards velocity of v0v_0from a height hh(both of these values are positive).
a) What is the point of maximum velocity?
b) What is the point of minimum acceleration?
I throw a ball horizontally off of a 30.0m-tall cliff. The ball hits Dave 20.0m from the base of the cliff. At what speed did I initially throw the ball? (Enter your answer in m/s, three sig figs)
Kinematics: Free fall
I drop a ball off a 100m building.
a) What is its velocity just before it hits the ground?
b) What is its average velocity?
Practice: Dropping a Basketball
I drop a basketball from the top of a tower that is 125 m tall. Find the time it takes for the ball to hit the ground. Enter your answer as a number of seconds with the appropriate number of sig figs. Use g = 9.81 m/s2.
Practice: Free Fall
A boy decided to estimate the height of a two-story building by throwing a stone up and measuring the time it takes for the stone to reach the top of the building. Assume he throws the stone from 1 meter above the ground, such that the stone stops and turns back just when it reached the top of the building.
A frog jumps by pushing off the ground, and accelerates very quickly while its feet are still on the ground. Its center of mass lifts 3.0 cm while it is pushing off. The frog reaches a height of 40 cm off the ground before falling again. What acceleration did it have while launching itself off the ground?
Practice: Measuring Height by Free Fall
Shannon decided to estimate the height of a two-storey building by throwing a stone up and measuring the time it takes for the stone to reach the top of the building. Assume she throws the stone from 1 meter above the ground, such that the stone stops and turns back just when it reached the top of the building.
a) If the stone takes 2 s2\ sto reach the top of the building, what is the initial velocity of the stone?
A gorilla that does not know how to drive, drives a car off a 10m cliff at speed of 20m/s. How far from the base of the cliff does the car land?
An object is thrown into the air. At the top of its trajectory, the vertical components of its acceleration and velocity are zero.
Practice question
A ball is thrown horizontally from top of building 55 m high strikes the ground at a point 35m from the building. What is the
(a) time to reach the ground
(b) initial speed of the ball
Practice: Free Fall
A boy decided to estimate the height of a 2-floor building by throwing a stone up and measure the time it takes for the stone to reach the top of the building. Assume he throws the stone such that the stone stops and turns back just when it reached the top of the building.
a) If the stone is thrown from 1m above the ground and it takes 2 s2\ sto reach the top of the building, what is the initial velocity of the stone?
b) What is the height of the building?
A 50-grams rubber ball is dropped from a height of 4 m. It hits the ground and bounces up, rising to height of 3.2 m. If the impact with ground only takes 0.05 second, what is the average force acting on the ball during the impact?
An apple is thrown upward from ground level at 20m/s. Determine the maximum height of the apple. Enter an answer with 1 decimal place.
Practice question : Free fall
An object is thrown vertically upwards with an initial velocity of 26.2 m/s. Determine the height (in meters) to which the object will rise above its initial height. Enter answer with 2 digits.