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

Projectile Motion

Two- and Three-Dimensional Kinematics (Calculus-Based)Previous Section
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Projectile Motion


When an object is launched in the air at an angle, its motion is two-dimensional and if it is only affected by gravity, this motion is known as Projectile Motion.








Wize Tip
Anytime we have two-dimensional kinematics problems (projectile motion or otherwise), the most important thing to realize is that the directions of motion do not affect each other; for example, an acceleration in the y-direction does not affect motion in the x-direction.

Think of it as solving a 1D kinematics problem two times in a row!

  • If we are given a velocity vvwith angle of θ\thetarespect to horizontal direction:

vx0=v0 cos⁡θv_{x0}=v_0\ \cos\theta

vy0=v0 sin⁡θv_{y0}=v_0\ \sin\theta

  • We can find time of flight to maximum height, maximum height and range of motion using the following equations respectively:

t=v0sinθgt = \dfrac{v_{0}sin\theta}{g}

h=v02sin2θ2gh = \dfrac{v^{2}_{0}sin^{2}\theta}{2g}

R=v02sin2θgR = \dfrac{v^{2}_{0}sin2\theta}{g}


Wize Tip
The only variable that does not change between the two dimensions is the time. This is the "link" that connects the two dimensions together.



Exam Tip
Note that for projectile motions, the only acceleration is due to gravity. Thus the x-component of the velocity does not change. The kinematics equations can be used as normal in the y-direction with a⃗=−g\vec{a}=-gwhen upward direction is defined to be positive.

Some Useful Notes about Projectile Motions

− Find the time to reach the ground from the y-component calculations.
− Find the horizontal distance (or range) from the x-component calculations.
− The velocity along the y-axis changes. vyo=0v_{yo}=0 at maximum height.
− Projectile motion is completely symmetric. For example, time to go up from a certain height equals time to come back down to that same height.
− Maximum range (horizontal distance) is obtained when the launch angle is 45°.



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Example: A Ball in Projectile Motion


A ball is shot at an angle of 30° from the horizontal. If the ball had an initial speed of 20m/s, how long was it in flight before it hit the ground?


In y-direction:
vo=20m/sLets calculate the time ball takes to reach the maximum heightv0y=20sin(30)=10m/say=−9.8m/s2t=?vfy=voy+ayt0=10+(9.8)tt=−10−9.8=1.02 sec    total time during the flight is T=2t=2sec\begin{array}{ll} v_o=20m/s\\ \text{Lets calculate the time ball takes to reach the maximum height}\\ v_{0y}=20sin(30)=10m/s\\ a_y=-9.8m/s^2\\ t=?\\ v_{fy}=v_{oy}+a_yt\\ 0=10+(9.8)t\\[3pt] t=\frac{-10}{-9.8}=1.02~sec\ \ \ \text{ total time during the flight is }\\ T = 2t = 2 sec \end{array}


A basketball player shoots a ball from 2-m height above the ground at angle of 30° respect to the horizontal direction toward a net. He scores in the net, which is 4.5 meters above the ground and 7.5 meters away from where the player stands.
What is the speed of the ball at which it was shot?
Extra Practice
Projectile motion
A man stands on the roof of a 15.0-m-tall building and throws a rock with a velocity of magnitude 30.0 m/s at an angle of 33.0∘33.0^{\circ} above the horizontal. You can ignore air resistance.
Calculate
(a) the maximum height above the roof reached by rock
Projectile motion
A man stands on the roof of a 15.0-m-tall building and throws a rock with a velocity of magnitude 30.0 m/s at an angle of 33.0∘33.0^{\circ} above the horizontal. You can ignore air resistance.
Calculate
(a) the maximum height above the roof reached by rock
Wyatt throws a ball upward at 50m/s at 75o above the horizontal. Frank, standing at the same place, does the same but at 15o above the horizontal. Which of the following is true?
I throw a ball horizontally off of a 30m-tall cliff. The ball hits Dave 20m from the base of the cliff. At what speed did I initially throw the ball?
Practice: Projectile motion
A man stands on the roof of a 15.0-m-tall building and throws a rock with a velocity of magnitude 30.0 m/s at an angle of 33.0∘33.0^{\circ} above the horizontal. You can ignore air resistance.
Calculate
(a) the maximum height above the roof reached by rock
Kinematics: Projectile
A cannonball is shot upwards at an angle θ\theta from the horizontal. Given that the ball had an initial speed of 20 m/s, and was in the air for one second, find the launch angle.
Enter your answer as a number of degrees, to three sig figs.
Kinematics: Projectile
I kick a soccer ball upward at an initial velocity of 25m/s at an angle of 45° from the horizontal. Determine a) the maximum height of the ball, b) the time it spent in the air, and c) the range (the horizontal distance travelled).
Practice: Battleships - calculation
You are on a battleship that fires cannonballs from a cannon. A launched cannonball reaches a max height of 30 meters, and strikes an enemy ship 15 meters away.
Find the initial velocity of the cannonball. State your answer two ways: one as a pair of components, and one as a magnitude and direction.
Practice: Projectile launch speed
It takes a projectile 0.29 seconds to fall from a height of 3.50 meters to a height of 1.50 meters on the downward portion of its trajectory, after initially being launched upwards. The total horizontal distance traveled, from launch to landing (which both occur at a height of zero), is 4.7 meters. What was the launch speed, and what was the launch angle?
Helping the Toronto Raptors
It's June 10, 2019, and the Toronto Raptors are playing their fifth game with a 3-1 series lead in the NBA Finals against the Golden State Warriors. You're watching the game in a packed bar. To everybody's disappointment, the Raptors are missing a lot of 3-point throws. Everybody is yelling at the TVs, but without any helpful information. You whip out a notebook to do some calculations.
Your calculation is for 3-point goals scored from the corners of the basketball court, from the 3-point line which is 6.71 meters away horizontally from the center of the hoop. The hoop is 3.05 meters above the ground. Kyle Lowry is 1.85 meters tall (assume he releases the ball from approximately the height of the top of his head). You estimate that the launch angle should be 75 degrees so that it falls through the hoop.
a) With what velocity should Lowry shoot? Give your answer in two forms:
You are launching a toy rocket and know it will launch at a speed of 125.0 mph (miles per hour). It hits the ground at a distance of 166.76 meters from where it started and was in the air for 3.1 seconds. (1 mile = 1.61 km)
a) How high up did it go?
b) What was the launch angle?
Projectile motion: Battleships
You are on a battleship with two cannons that can fire at the same time, at different angles. There are two enemy ships nearby and you launch cannonballs at each one. The first cannonball reaches a max height of 30 meters, aimed at the enemy ship 15 meters away. The second cannonball reaches a max height of 15 meters, aimed at the enemy ship 30 meters away.
a) Draw a sketch representing the trajectories of the cannonballs.
b) Which cannonball strikes its target first? Or, do they strike their targets at the same time? Explain your answer.
A football is punted at 45.0o , and it travels 60.0m horizontally, assuming no air drag. What was the initial speed?
Wyatt throws a ball upward at 50m/s at 75o above the horizontal. Frank, standing at the same place, does the same but at 15o above the horizontal. Which of the following is true?
Problems on Projectile Motion
Your angry neighbour hates your new lawn gnome, sitting 100m the base of his house. So he stands on the roof of his 15.0-m-tall house and throws a rock with a velocity of magnitude 30.0 m/s at an angle of 33.0∘33.0^{\circ} above the horizontal. You can ignore air resistance.
Linda, while walking to work, got pooped on by a pigeon. In a fit of rage, she throws a nearby rock upward at the bird. Her anger has given her a boost of adrenaline, and she hurls the rock 70.0m/s at an angle of 75o. The rock hits the pigeon at the top of it's trajectory. What was the magnitude displacement of the bird relative to Linda (in metres)?
Alice and Bob throw pennies into the fountain at the exact same time. Bob is a couple metres in front of Alice. Whose penny landed first?
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?
We know that v = 215 km/h, h = 200 m.)
2D kinematics - soccerball
Christian kicks a soccerball at an angle of 53.0o to the horizontal, with an intial speed of 10.0m/s. What is the maximum height of the ball?
Shooting two bullets concept
Alice and Bob stand back to back. Alice shoots her gun parallel to the ground, and Bob shoots it point up a little bit. Which of the following is true?
A ball is launched into the air by a cannon angled 57.0º above the horizon, at a speed of 278 m/s.
a) What is the horizontal component of the ball’s velocity? Enter answer in m/s
b) At what time will the ball reach its maximum height? Enter answer in sec
Projectile motion
During projectile motion
(a) what is the value of velocity at the maximum height of an object?
(b) What is the horizontal component of acceleration?
Practice: Projectile Motion
I kick a soccer ball with an initial speed of 25m/s at an angle of 45° from the horizontal. Determine the maximum height of the ball, the time it spent in the air, and the range.
A boy threw an apple with an initial speed vi and angle of 30 degrees with respect to the horizontal surface from the top of a 5-m tall tree (Ignore any air drag force).
a) If it takes 4s for the ball to hit the ground, what should be the initial speed?
b) What is the speed of the apple at which it hits the ground?
I throw a ball horizontally off of a 30m-tall cliff. The ball hits Dave 20m from the base of the cliff. At what speed did I initially throw the ball?
A golf ball is hit with an initial speed of 30m/s at an angle of 50 degrees above the horizontal. There is a wall with 4m of height at a distance of 20 m.
a) Does the ball hit the wall?
b) What should be the angle of hitting such that the ball just touches the base of the wall?
A man is standing on top of a building with the height of 15m when he throws a stone with initial speed of 20m/s with an angle of 60 degrees respect to the horizontal line. Find the time it hits the ground and the distance between the point it hits the ground and base of the building.
A daring 510 N swimmer dives off a cliff with a running horizontal leap. What must her minimum speed be just as she leaves the top of the cliff so that she will miss the ledge at the bottom, which is 1.75 m wide and 9.00 m below the top of the cliff?
A ball is shot at an angle of 30° from the horizontal. If the ball had an initial speed of 20m/s, how long was it in flight before it hit the ground?
Projectile motion
An object is launched at a velocity of 20 m/s in a direction making an angle of 25° upward with the horizontal. a) What is the maximum height reached by the object? b) What is the total flight time (between launch and touching the ground) of the object? c) What is the horizontal range (maximum x above ground) of the object? d) What is the magnitude of the velocity of the object just before it hits the ground?
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 in sec
(b) initial speed of the ball in m/s
A ball is launched from a height of 200m at 100m/s at 41o above the horizontal. Find: a) The maximum height of the ball b) The time of the flight (TOF) of the ball c) The maximum range of the ball d) The final velocity of the ball (relative to the ground) just before it lands