# 5.5 Projectile motion types (application)

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Solving problems is an essential part of the understanding process.

Questions and their answers are presented here in the module text format as if it were an extension of the treatment of the topic. The idea is to provide a verbose explanation, detailing the application of theory. Solution presented is, therefore, treated as the part of the understanding process – not merely a Q/A session. The emphasis is to enforce ideas and concepts, which can not be completely absorbed unless they are put to real time situation.

## Hints on solving problems

1: Identify projectile motion types. The possible variants are :

• Projectile is thrown in horizontal direction. In this case, initial vertical component of velocity is zero. Consider horizontal direction as positive x-direction and vertically downward direction as positive y-direction.
• Projectile is thrown above horizontal level. The projectile first goes up and then comes down below the level of projection
• Projectile is thrown below horizontal level. Consider horizontal direction as positive x-direction and vertically downward direction as positive y-direction.

2: We can not use standard equations of time of flight, maximum height and horizontal range. We need to analyze the problem in vertical direction for time of flight and maximum height. Remember that determination of horizontal range will involve analysis in both vertical (for time of flight) and horizontal (for the horizontal range) directions.

3: However, if problem has information about motion in horizontal direction, then it is always advantageous to analyze motion in horizontal direction.

## Representative problems and their solutions

We discuss problems, which highlight certain aspects of the study leading to the projectile motion types. The questions are categorized in terms of the characterizing features of the subject matter :

• Time of flight
• Range of flight
• Initial velocity
• Final velocity

## Time of flight

Problem : A ball from a tower of height 30 m is projected down at an angle of 30° from the horizontal with a speed of 10 m/s. How long does ball take to reach the ground? (consider g = 10 $\phantom{\rule{2pt}{0ex}}m/{s}^{2}$ )

Solution : Here, we consider a reference system whose origin coincides with the point of projection. Further, we consider that the downward direction is positive y - direction.

Motion in vertical direction :

Here, $\phantom{\rule{2pt}{0ex}}{u}_{y}=u\mathrm{sin}\theta =10\mathrm{sin}{30}^{0}=5\phantom{\rule{2pt}{0ex}}m/s;y=30\phantom{\rule{2pt}{0ex}}m;$ . Using $\phantom{\rule{2pt}{0ex}}y={u}_{y}t+\frac{1}{2}{a}_{y}{t}^{2}$ , we have :

$\begin{array}{l}⇒\mathrm{30}=5t+\frac{1}{2}10{t}^{2}\\ ⇒{t}^{2}+t-6=0\\ ⇒t\left(t+3\right)-2\left(t+3\right)=0\\ ⇒t=-3\phantom{\rule{2pt}{0ex}}s\phantom{\rule{2pt}{0ex}}\mathrm{or}\phantom{\rule{2pt}{0ex}}t=2\phantom{\rule{2pt}{0ex}}s\end{array}$

Neglecting negative value of time, t = 2 s

## Range of flight

Problem : A ball is thrown from a tower of height “h” in the horizontal direction at a speed “u”. Find the horizontal range of the projectile.

Solution : Here, we consider a reference system whose origin coincides with the point of projection. we consider that the downward direction is positive y - direction.

$\begin{array}{l}⇒x=R={u}_{x}T=uT\end{array}$

Motion in the vertical direction :

Here, $\phantom{\rule{2pt}{0ex}}{u}_{y}=0\phantom{\rule{2pt}{0ex}}$ and t = T (total time of flight)

$\begin{array}{l}⇒h=\frac{1}{2}g{T}^{2}\\ ⇒T=\sqrt{\left(\frac{2h}{g}\right)}\end{array}$

Putting expression of total time of flight in the expression for horizontal range, we have :

$\begin{array}{l}⇒R=u\sqrt{\left(\frac{2h}{g}\right)}\end{array}$

#### Questions & Answers

List the application of projectile
Luther Reply
How can we take advantage of our knowledge about motion?
Kenneth Reply
pls explain what is dimension of 1in length and -1 in time ,what's is there difference between them
Mercy Reply
what are scalars
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show that 1w= 10^7ergs^-1
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what's lamin's theorems and it's mathematics representative
Yusuf Reply
if the wavelength is double,what is the frequency of the wave
Ekanem Reply
What are the system of units
Jonah Reply
A stone propelled from a catapult with a speed of 50ms-1 attains a height of 100m. Calculate the time of flight, calculate the angle of projection, calculate the range attained
Samson Reply
58asagravitasnal firce
Amar
water boil at 100 and why
isaac Reply
what is upper limit of speed
Riya Reply
what temperature is 0 k
Riya
0k is the lower limit of the themordynamic scale which is equalt to -273 In celcius scale
Mustapha
How MKS system is the subset of SI system?
Clash Reply
which colour has the shortest wavelength in the white light spectrum
Mustapha Reply
how do we add
Jennifer Reply
if x=a-b, a=5.8cm b=3.22 cm find percentage error in x
Abhyanshu Reply
x=5.8-3.22 x=2.58
sajjad

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Source:  OpenStax, Physics for k-12. OpenStax CNX. Sep 07, 2009 Download for free at http://cnx.org/content/col10322/1.175
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