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Any motion having the following conditions is called the projectile motion

(i) Follows a parabolic path (trajectory)

(ii) Moves under the influence of gravity.

The __projectile motion formulas__ applying to solve two-dimensional projectile motion problems are as follows \begin{gather*} x=(v_0\cos\theta)t+x_0\\\\y=-\frac 12 gt^2+(v_0\sin\theta)t+y_0\\\\ v_y=v_0\sin\theta-gt\\\\v_y^2-(v_0\sin\theta)^2=-2g(y-y_0)\end{gather*} In the next section, we want to know how to solve projectile motion problems using these kinematic equations.

**Problem (1): A person kicks a ball with an initial velocity $15\,{\rm m/s}$ at an angle 37° above the horizontal (neglect the air resistance). Find
(a) the total time the ball is in the air.
(b) the horizontal distance traveled by the ball**

**Solution**: To solve any projectile problems, first of all, adopt a coordinate system and draw its projectile path and put the initial and final positions, velocities.

By doing this, you can able to solve the relevant projectile equations easily.

Therefore, we choose the origin of the coordinate system to be at the throwing point, $x_0=0, y_0=0$.

**(a)** Here, the time between throwing and striking the ground is wanted.

In effect, the projectiles have two independent motions, one is in the horizontal direction with uniform motion at constant velocity i.e. $a_x=0$, and the other is in the vertical direction under the effect of gravity with $a_y=-g$.

The kinematic equations that describe the horizontal and vertical distances are as follows \begin{gather*} x=x_0+(\underbrace{v_0\cos \theta}_{v_{0x}})t \\ y=-\frac 12 gt^2+(\underbrace{v_0\sin \theta}_{v_{0y}})t+y_0\end{gather*} By substituting the coordinates of the initial and final points into the vertical equation, we can find the total time the ball is in the air.

Setting $y=0$ in the second equation, we have \begin{align*} y&=-\frac 12 gt^2+(v_0\sin \theta)t+y_0\\\\ 0&=-\frac 12 (9.8)t^2+(15)\sin 37^\circ\,t+0 \end{align*} By rearranging the above expression, we can get two solutions for $t$: \begin{gather*} t_1=0 \\\\ t_2=\frac{2\times 15\sin37^\circ}{9.8}=1.84\,{\rm s}\end{gather*} The first time is for the starting moment and the second is the total time the ball was in the air.

**(b)** As mentioned above, the projectile motion is made up of two independent motions with different positions, velocities, and accelerations which two distinct kinematic equations describe those motions.

Between any two desired points in the projectile path, the time needed to move horizontally to reach a specific point is the same time needed to falls vertically to that point.

This is an important observation in solving the projectile motion problems.

Therefore, time is the only common quantity in the horizontal and vertical motions of a projectile. In this problem, the time obtained in part (a) can be substituted in the horizontal kinematic equation, to find the distance traveled as below \begin{align*} x&=x_0+(v_0\cos \theta)t \\ &=0+(15)\cos 37^\circ\,(1.84) \\ &=22.08\quad {\rm m}\end{align*}

**Problem (2): A ball is thrown into the air at an angle of $60^\circ$ above the horizontal with an initial velocity of $40\,{\rm m/s}$ from a 50-m-high building. Find
(a) The time to reach the top of its path.
(b) The maximum height the ball is reached from the base of the building.**

**Solution**: In the previous question, we find that the projectile is a motion composed of two vertical and horizontal motions.

We learned about how to find distances in both directions using relevant kinematic equations.

There are also another set of kinematic equations which discuss about the velocities in vertical and horizontal directions as following \begin{gather*} v_x=v_{0x}=v_0\cos\theta \\ v_y=v_{0y}-gt=v_0\sin\theta-gt\end{gather*} As you can see, the horizontal component of the velocity, $v_x$, is constant throughout the motion but the vertical component varies with time.

As an important note, keep in mind that in the problems about projectile motions, at the highest point of the trajectory the vertical component of the velocity is always zero i.e. $v_y=0$.

To solve the first part of the problem, specify two points as initial and final points, then solve the relevant kinematic equations between those points.

Here, setting $v_y=0$ in the second equation and solving for the unknown time $t$, we have \begin{align*} v_y&=v_{0y}-gt=v_0\sin\theta-gt\\ 0&=(40)(\sin 60^\circ)-(9.8)(t) \\ \Rightarrow t&=3.53\quad {\rm s}\end{align*} Thus, the time taken to reach the maximum height of the trajectory (path) is 3.53 seconds from moment of the launch. We call this as maximum time $t_{max}$.

**(b)** Let the origin be at the throwing point so $x_0=y_0=0$ in the kinematic equations. In this part, vertical distance traveled to the maximum point is requested.

By substituting $t_{max}$ into the vertical distance projectile equation, we can find the maximum height as below \begin{align*} y-y_0&=-\frac 12 gt^2 +(v_0 \sin \theta)t\\ \\ y_{max}&=-\frac 12 (9.8)(3.53)^2+(40\times \sin 60^\circ)(3.53)\\\\ &=61.22\quad {\rm m}\end{align*} Therefore, the maximum height the ball is reached from the base of the building is \[H=50+y_{max}=111.22\quad {\rm m}\]

*For further reading about uniform motion along the horizontal direction, read speed, velocity, and acceleration problems.*

**Problem (3): A projectile is fired horizontally at a speed of 8 m/s from an 80-m-high cliff. Find
(a) The velocity just before the projectile hits the ground.
(b) The angle of impact.**

**Solution**: In this problem, the angle of the projectile is zero because it is fired horizontally.

Velocity at each point of a projectile trajectory (path) is obtained by the following formula \[v=\sqrt{v_x^2+v_y^2}\] where $v_x$ and $v_y$ are the horizontal and vertical components of the projectile's velocity at any instant of time.

**(a)** Recall that the horizontal component of projectile's velocity is always constant and, for this problem, is found as \begin{align*} v_x&=v_0\cos\theta\\&=8\times \cos 0^\circ\\&=8\quad {\rm m/s}\end{align*} To find the vertical component of the projectile velocity at any moment, $v_y=v_0\sin\theta-gt$, we should find the time taken to that point.

In this problem, that point is located just before striking the ground whose coordinate is $y=-80\,{\rm m}, x=?$. Because it is below the origin, which is assumed to be at the firing point, we inserted a minus sign.

Because __displacement __in the vertical direction is known so we can use the projectile formula for vertical distance.

By setting $y=-80$ into it and solving for time $t$ needed the projectile reaches the ground, we get \begin{align*} y&=-\frac 12 gt^2+(v_0\sin\theta)t\\\\-80&=-\frac 12 (9.8)t^2+(8\times \sin 0^\circ)t\\\\\Rightarrow t&=\sqrt{\frac{2(80)}{9.8}}\\\\&=2.86\quad {\rm s}\end{align*} Now insert this time into the y-component of the projectiles' velocity to find the $v_y$ just before hitting the ground \begin{align*} v_y&=v_0\sin\theta-gt\\ &=8\sin 0^\circ-(9.8)(2.86)\\&=-28\quad {\rm m/s}\end{align*} Now that both components of the velocity are available, we can compute its magnitude as below \begin{align*} v&=\sqrt{v_x^2+v_y^2}\\\\&=\sqrt{8^2+(-28)^2}\\\\&=29.1\quad {\rm m/s}\end{align*} Therefore, the projectile is hit the ground at a speed of 29.1 m/s.

(b) At any instant of time, the velocity of projectile makes some angle with horizontal whose magnitude is obtained as the following formula \[\alpha=\tan^{-1}\left(\frac{v_y}{v_x}\right)\] Substituting the above values into this formula we get \[\alpha =\tan^{-1}\left(\frac{-28}{8}\right)=-74^\circ\] Therefore, the projectile is hit the ground at an angle of 74° below the horizontal.

To understand better the components of a vector refer to the following:

__Vectors problems with solutions__

**Problem (4): From a cliff of 100-m high, a ball is kicked at $30^\circ$ above the horizontal with a speed of $20\,{\rm m/s}$. How far from the base of the cliff the ball hit the ground? (Assume $g=10\,{\rm m/s^2}$). **

**Solution:** Again, similar to any projectile motion problem, we first select a coordinate system and then draw the path of the projectile as the figure below,

We choose the origin to be at the kicking point above the cliff so setting $x_0=y_0=0$ in the kinematic equations.

The coordinate of the hitting point to the ground is $y=-100\,{\rm m} , x = ?$. A negative is inserted because the final point is below the origin.

Now, we find the common quantity in the projectile motions which is the time between the initial and final points.

To find the total time the ball was in the air, we can use the vertical equation and solve for the unknown $t$ as follows \begin{align*} y&=-\frac 12 gt^2 +(v_0\sin \theta)t \\\\ -100&=-\frac 12 (10) t^2+(20\sin 30^\circ)t\\\\&\Rightarrow \ t^2-2t-20=0 \end{align*} The solutions of a quadratic equation $at^2+bt+c=0$ are found by the formula below \[t=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\] By matching the above constant coefficients with out quadratic equation, we can find the total time as below \[t=\frac{-(-2)\pm\sqrt{(-2)^2-4(1)(-20)}}{2(1)}\] After simplifying, two solutions are obtained, $t_1=5.6\,{\rm s}$ and $t_2=-3.6\,{\rm s}$.

There is obvious that time can not be negative in physics so the acceptable answer is $t_1$.

This is the time it takes the ball to travel in the vertical direction. On the other hand, it is also the time it takes the ball to travel the horizontal distance between kicking and hitting points.

We insert this time into the horizontal equation to find the horizontal distance traveled which is known as range of projectile. \begin{align*} x&=(v_0\cos \theta)t\\ &=(20)(\cos 30^\circ)(5.6)\\&=97\quad {\rm m}\end{align*}

**Problem (5): A cannonball is fired from a cliff with a speed of 800 m/s at an angle of 30° below the horizontal. How long will it take to reach 150 m below the firing point? **

**Solution:** First, choose the origin to be at the firing point so $x_0=y_0=0$. Now, list the known values below

(i) Projectile's initial velocity = 800 m/s

(ii) Angle of projectile : $-30^\circ$, a minus is due to below the horizontal.

(iii) y-coordinate of the final point, 150 m below the origin, $y=-150\,{\rm m}$.

In this problem, the length of time it takes for the cannonball to reach 100 m below the starting point is required.

Since the displacement to that point is known, we apply the vertical displacement projectile formula to find the needed time as below \begin{align*} y-y_0&=-\frac 12 gt^2+(v_0\sin\theta)t\\\\-150&=-\frac 12 (9.8)t^2+(800)\sin(-30^\circ)t\\\\ \Rightarrow & \ 4.9t^2+400t-150=0\end{align*} The above quadratic equation has two solutions, $t_1=0.37\,{\rm s}$ and $t_2=-82\,{\rm s}$. It is obvious that the second time is not acceptable.

Therefore, the cannonball takes 0.37 seconds to reach 150 meters below the firing point.

**Problem (6): Someone throws a stone into the air from ground level. The horizontal component of velocity is 25 m/s and it takes 3 s to stone come back to the same height as before. Find
(a) The range of the stone.
(b) The initial vertical component of velocity
(c) The angle of projection. **

**Solution:** The known value is

(i) The initial horizontal component of velocity, $v_{0x}=25\,{\rm m/s}$.

(ii) The time between initial and final points, which are at the same level, $t=3\,{\rm s}$.

(a) Range of projectile motion is defined as the horizontal distance between launch point and impact at the same elevation.

Because the horizontal motion in projectiles is a motion with constant velocity so distance traveled in this direction is obtained as $x=v_{0x}t$, where $v_{0x}$ is the initial component of the velocity.

If you put the total time the projectile is in the air into this formula, you get the range of the projectile.

In this problem, the stone is thrown from the ground level and after 3 s reaches the same height. Thus, this is the total time of the projectile.

Hence, the range of the stone is found as below \begin{align*} x&=v_{0x}t\\&=(25)(3)\\&=75\,{\rm m}\end{align*}

(b) Initial vertical component of projectile's velocity appear in two equations, $v_y=v_{0y}-gt$ and $y-y_0=-\frac 12 gt^2+v_{0y}t$.

Using the second formula is more straightforward because the stone reaches the same height so its vertical displacement between initial and final points is zero i.e. $y-y_0=0$. Setting this into the vertical distance projectile equation, we get \begin{align*} y-y_0&=-\frac 12 gt^2+v_{0y}t\\\\ 0&=-\frac 12 (9.8)(3)^2+v_{0y}(3) \\\\ \Rightarrow v_{0y}&=14.7\quad{\rm m/s}\end{align*} To use the first formula, we need some extra facts about projectile motion in the absence of air resistance as below

(i) The vertical velocity is zero at the highest point of the path of the projectile i.e. $v_y=0$.

(ii) If the projectile lands at the same elevation from which it was launched, then the time it takes to reach the highest point of the trajectory is half of the total time between initial and final points.

The second note in the absence of the air resistance is only valid.

In this problem, the total flight time is 3 s because air resistance is negligible so 1.5 seconds takes the stone to reach the maximum height of its path.

Therefore, using second equation we can find $v_{0y}$ as below \begin{align*} v_y&=v_{0y}-gt\\0&=v_{0y}-(9.8)(1.5) \\\Rightarrow v_{0y}&=14.7\quad {\rm m/s}\end{align*}

(c) The projection angle is the angle at which the projectile is thrown into the air and performs a two-dimensional motion.

Once the components of the initial velocity are available, using trigonometry we can find the angle of projection as below \begin{align*} \theta&=\tan^{-1}\left(\frac{v_{0y}}{v_{0x}}\right)\\\\&=\tan^{-1}\left(\frac{14.7}{25}\right)\\\\&=+30.45^\circ\end{align*} Therefore, the stone is thrown into the air at an angle of about 30° above the horizontal.

**Problem (7): A ball is thrown at an angle of 60° with an initial speed of 200 m/s. (Neglect the air resistance)
(a) How long is the ball in the air?
(b) Find the maximum horizontal distance traveled by the ball.
(c) What is the maximum height reached by the ball?**

**Solution:** We choose the origin to be the initial position of the ball so that $x_0=y_0=0$. The given data is

(i) The projection angle : $60^\circ$.

(ii) Initial speed : $v_0=200\,{\rm m/s}$.

(a) The initial and final points of the ball are at the same level i.e. $y-y_0=0$.

Thus, the total time the ball is in the air is found by setting $y=0$ in the projectile equation $y=-1/2 gt^2+v_{0y}t$ and solving for time $t$ as below \begin{align*} y&=-\frac 12 gt^2+(v_0\sin\theta)t\\\\0&=-\frac 12 (9.8)t^2+(200)(\sin 60^\circ)t\\\\\Rightarrow & (-4.9t+100\sqrt{3})t=0 \end{align*} The above expression has two solutions for $t$. One is the initial time, $t_1=0$, and the other is computed as $t_2=35.4\,{\rm s}$.

Hence, the ball is in the air for about 35 s.

(b) The horizontal distance is called the range of the projectile. By inserting the above time (total flight time) into the horizontal distance projectile equation $x=v_{0x}t$, we can find the desired distance traveled. \begin{align*} x&=(v_0\cos\theta)t\\&=(200)(\cos 60^\circ)(35.4)\\&=3540 \quad {\rm m}\end{align*} Therefore, the ball is hit the ground 3540 meters away from the throwing point.

(c) Using projectile equation $v_y^2-v_{0y}^2=-2g(y-y_0)$, setting $v_y=0$ at the highest point of the path and solving for vertical distance $y$, the maximum height is found as following \begin{align*} v_y^2-v_{0y}^2&=-2g(y-y_0)\\0-(200\sin 60^\circ)^2&=-2(9.8)y\\\Rightarrow y&=1531\quad {\rm m}\end{align*} Another method: As mentioned above, the ball is hit the ground at the same level as before so by having total flight time and halving it, we can find the time it takes the ball reaches the highest point of its trajectory.

Therefore, setting the half of the total flight time in the following projectile kinematic formula and solving for $y$, we can find the maximum height as \begin{align*} y-y_0&=-\frac 12 gt^2+(v_0\sin\theta)t\\\\y&=-\frac 12 (9.8)(17.7)^2+(200\sin60^\circ)(17.7)\\\\&=1531\quad {\rm m}\end{align*} Hence, the ball reaches 1531 meters above the launch point.

**Problem (8): What are the horizontal range and maximum height of a bullet fired with a speed of 20 m/s at 30° above the horizontal?**

**Solution**: first find the total flight time, then insert it into the horizontal displacement projectile equation $x=v_{0x}t$ to find the range of the bullet.

Because the bullet lands at the same level as original so setting its vertical displacement is zero, $y-y_0=0$, in the following projectile formula we can find the total flight time \begin{align*} y-y_0&=-\frac 12 gt^2+(v_0\sin\theta)t\\\\0&=-\frac 12 (9.8)t^2+(20)(\sin30^\circ)t\\\\ \Rightarrow & (-4.9t+10)t=0\end{align*} Solving for time $t$ gives two solutions, one is the initial time $t_1=0$, and the other is $t_2=1.02\,{\rm s}$. Thus the total time of flight is 2.04 s.

Therefore, the maximum horizontal distance traveled by the bullet, which is defined as the range of the projectile, is calculated as \begin{align*} x&=(v_0\cos\theta)t\\&=(20\cos 30^\circ)(2.04)\\&=35.3\quad {\rm m}\end{align*} Hence, the bullet is landed about 17 m away from the launch point.

Because the air resistance is negligible and the bullet lands at the same height as the original, so the time it takes to reach the highest point of its path, is always half the total flight time.

On the other hand, recall that the vertical component of velocity at the maximum height is always zero i.e. $v_y=0$. By inserting these two note into the following projectile equation, we have \begin{align*} y&=-\frac 12 gt^2+(v_0\sin\theta)t\\\\&=-\frac 12 (9.8)(1.02)^2+(20\sin 30^\circ)(1.02)\\\\&=5.1\quad {\rm m}\end{align*} We could also simply use the kinematic equation $v_y^2-v_{0y}^2=-2g(y-y_0)$, to find the maximum height as below \begin{align*} v_y^2-(v_0 \sin \theta)^2 &=-2g(y-y_0)\\ 0-(20\sin 30^\circ)^2 &=-2(9.8)H \\ \Rightarrow H&= 5.1\quad {\rm m} \end {align*} I think the second method is much simpler.

**Author**: Ali Nemati

**Date Published:** 6/15/2021

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