10Motion of a body dropped at an angle to the horizon

Theoretical material

Projectile refers to an object that is in flight after being thrown or projected. In a projectile motion, the only acceleration acting is in the vertical direction which is acceleration due to gravity (g). Equations of motion, therefore, can be applied separately in X-axis and Y-axis to find the unknown parameters.

Some examples of Projectile Motion are Football, A baseball, A cricket ball, or any other object. The projectile motion consists of two parts – one is the horizontal motion of no acceleration and the other vertical motion of constant acceleration due to gravity. The projectile motion is always in the form of a parabola which is represented as:

y = ax + bx²

When a particle is projected obliquely near the earth’s surface, it moves simultaneously in the direction of horizontal and vertical. The motion of such a particle is called Projectile Motion. In the above diagram, where a particle is projected at an angle θ, with an initial velocity u. For this particular case, we will calculate the following:

•      The time is taken to reach point A from O

•      The horizontal distance covered (OA)

•      The maximum height reached during the motion.

•      The velocity at any time “ t “ during the motion.

If any object is thrown with the velocity u, making an angle Θ from horizontal, then the horizontal component of initial velocity = u cos Θ and the vertical component of initial velocity = u sin Θ. The horizontal component of velocity (u cos Θ) remains the same during the whole journey as herein, no acceleration is acting horizontally.

The vertical component of velocity (u sin Θ) gradually decrease and at the highest point of the path becomes 0. The velocity of the body at the highest point is u cos Θ in the horizontal direction. However, the angle between the velocity and acceleration is 90 degree.

Additional differentiated tasks and problems

1)   A skier leaves a ramp with a velocity at an angle of 370 with the horizontal and 5 s later he lands on the ground, as shoen in the figure.

•      If his velocity components, 2 s after he leaves the ramp, are as shown in the figure, find

a)      The initial velocity  of the skier at the moment of leaving the ramp.

b)      The skier’s final velocity at the moment of landing on the ground.

c)      The time the skier takes to reach his maximum height.

d)     

2)      A ball of 600 grams is kicked at an angle of 35° with the ground with an initial velocity V₀.

a) What is the initial velocity V₀ of the ball if its kinetic energy is 22 Joules when its height is maximum?

b) What is the maximum height reached by the ball

3) A projectile starting from ground hits a target on the ground located at a distance of 1000 meters after 40 seconds.

a) What is the size of the angle θ?

b) At what initial velocity was the projectile launched?

4) Two balls A and B of masses 100 grams and 300 grams respectively are pushed horizontally from a table of height 3 meters. Ball has is pushed so that its initial velocity is 10 m/s and ball B is pushed so that its initial velocity is 15 m/s.

a) Find the time it takes each ball to hit the ground.

b) What is the difference in the distance between the points of impact of the two balls on the ground?

Useful links and literature

1)      https://www.toppr.com/guides/physics/motion-in-a-plane/projectile-motion/

2)      https://www.khanacademy.org/science/physics/two-dimensional-motion/two-dimensional-projectile-mot/a/what-is-2d-projectile-motion

3)      https://phys.libretexts.org/Bookshelves/University_Physics/Book%3A_Physics_(Boundless)/3%3A_Two-Dimensional_Kinematics/3.3%3A_Projectile_Motion

4)      https://www.physicsclassroom.com/class/vectors/Lesson-2/What-is-a-Projectile

5)      http://www.problemsphysics.com/physics-calculators-solvers/projectile-motion-calculator-and-solver.html

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