Projectile Motion Example: Finding Angles of Projection
Learn how to find two possible angles for the projection of a projectile based on horizontal and vertical distances traveled, with step-by-step calculations and trigonometric identities.
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Projectiles passing through a point A worked example Question: A projectile is fired on horizontal ground with an initial speed of 10 7 m/s at and angle to the horizontal. When the projectile has travelled 10 m horizontally it is at a height of 18 5 m. Find two possible angles of projection.
Some background thoughts to this question: It can be helpful to consider the projectile as passing through a specific point ? ?,? ? during its motion. In this example the horizontal distance is 10 and the vertical distance is 18 5. As a co- ordinate point, this is represented by 10,18 5 . When solving these questions, we can: a) establish an expression for ? from ? ?. b) fill this value of ? into the expression for ? ?, and solve the resulting quadratic equation.
Resolving the initial speed of 10 7 into a vector we get: ? = 10 7???? ? + 10 7???? ? Taking ? ?= 10 and noting there is no horizontal acceleration so the formula ? = ?? +1 2??2simplifies to ? = ?? we get: 10 = 10 7cos?? This allows us to find an expression for time: ? = 10 10 7 cos ? As we know the vertical displacement is 18 5, we should fill this into our expression for ? ?, noting that the acceleration in the formula ? = ?? +1 given by gravity, represented by ?. 10 7cos? ? At this stage, we need to be aware of two trigonometric identities: 2??2is 10 100 18 5 = 10 7sin? 2 100 7 ???2? i. ???? ????= ???? 1 ii. ???2?= (1 + ???2?) Using these, and simplifying some of the numbers, the above equation simplifies to: 7 10(1 + ???2?) now apply a common denominator of 10: 18 5 = 10????
185 = 100???? 7 7???2? 7???2? 100???? + 192 = 0 7???? 16 ???? 12 = 0 7???? 16 = 0 ???? 12 = 0 7???? = 16 ???? = 12 ???? =16 ???? = 12 7 16 7 ? = ??? 1 ? = ??? 1(12)
Some extended thoughts. This type of question can appear in different guises if a projectile just clears a wall of a certain height a certain distance from the point of projection, we can assume the distance of the wall from the point of projection and its height to be a co-ordinate point ? ?,? ? and proceed as outlined in the worked example. Whilst ???? frequently in Applied Maths and regular Maths that you will become quite familiar with it. ????= ???? is in the Formulae and Tables Book (Pg. 13), it is used 1 However, book so you should learn it specifically for these questions. 1+???2?= ???2? is not readily available in the Formulae and Tables
