# 5.7 Projectile motion on an incline (application)

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Solving problems is an essential part of the understanding

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 for solving problems

Problems based on projectile motion over an incline are slightly difficult. The analysis is complicated mainly because there are multitudes of approaches available. First there is issue of coordinates, then we might face the conflict to either use derived formula or analyze motion independently in component directions and so on. We also need to handle motion up and down the incline in an appropriate manner. However, solutions get easier if we have the insight into the working with new set of coordinate system and develop ability to assign appropriate values of accelerations, angles and component velocities etc.

Here, we present a simple set of guidelines in a very general way :

1: Analyze motion independently along the selected coordinates. Avoid using derived formula to the extent possible.

2: Make note of information given in the question like angles etc., which might render certain component of velocity zero in certain direction.

3: If range of the projectile is given, we may try the trigonometric ratio of the incline itself to get the answer.

4: If we use coordinate system along incline and in the direction perpendicular to it, then always remember that component motion along both incline and in the direction perpendicular to it are accelerated motions. Ensure that we use appropriate components of acceleration in the equations of motion.

## Representative problems and their solutions

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

• Range of the flight
• Angle of projection
• Final Speed of the projectile
• Elastic collision with the incline
• Projectile motion on double inclines

## Range of the flight

Problem : A projectile is thrown with a speed "u" at an angle 60° over an incline of 30°. If the time of flight of the projectile is “T”, then find the range of the flight.

Solution : We can see here that time of flight is already given. We can find range considering projectile motion in the coordinates of horizontal and vertical axes. The range of the projectile “R” is obtained by using trigonometric ratio in triangle OAB. The range is related to horizontal base “OB” as :

$\mathrm{cos}{30}^{0}=\frac{OB}{OA}$

$R=OA=\frac{OB}{\mathrm{cos}{30}^{0}}=OB\mathrm{sec}{30}^{0}$

Now, we can find OB by considering motion in horizontal direction :

$⇒R=OB={u}_{x}T=u\mathrm{cos}{60}^{0}T=\frac{uT}{2}$

Thus, the range of the projectile, OA, is :

$⇒R=OA=\frac{uT\mathrm{sec}{30}^{0}}{2}=\frac{uT}{\sqrt{3}}$

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