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| Oblique Projectile |
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| The figure above shows a projectile thrown with a velocity v at an angle q with the horizontal. The velocity v is resolved into its two rectangular components, viz., |
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 v Cosq along X-axis which is constant, since, the force of gravity in the horizontal direction is zero. |
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 v sinq in the vertically upward direction, decelerating with time, since, it is acted upon by the force of gravity in the vertically downward direction. |
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| Therefore, the equations of motion of the projectile for the horizontal direction are merely the equations of motion in a straight line. If x is the distance covered in the horizontal direction in time t, then |
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| The motion in the vertical direction is under the influence of the force of gravity, therefore, its velocity vy or v sinq, steadily decreases with time and becomes zero at a certain height which is the maximum height attained by the projectile. After this, the projectile reverses its direction and returns to the ground and strikes the ground with a speed v, which is the same as the initial speed of the projectile. |
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| Let y be the vertical distance covered by the projectile in time t. Considering the vertical motion of the projectile |
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| However from equation (1), |
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 Equation (2) becomes, |
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| This is the equation of a parabola, as the general equation of a parabola is y = ax - bx2. |
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| It is clear from equation (3) that the trajectory is completely known if v and q are known. It is also important to remember that equation (3)
is value only if q lies between 0 and
p/2. |
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| In the figure above, the projectile is at p at time t. Let v be the resultant velocity of the projectile at time t. This velocity is along the tangent to the trajectory at point p. Again, the motion in the horizontal direction is uniform. Therefore, the horizontal component of the velocity will remain constant. |
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| To calculate the vertical component of the velocity vy, which is under the influence of the force of gravity acting in the opposite direction, we gather all our data that is, |
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| We have, |
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| Applying the law of parallelogram of vectors, |
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| Since, |
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| This equation gives the magnitude of the resultant velocity of the projectile at any time. |
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| The direction of the projectile is given by the angle equation, which the resultant velocity v makes with the horizontal direction. |
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