When a projectile is launched with a non-zero horizontal velocity, its trajectory takes on the shape of a parabola instead of just the linear trajectory it had when released either from rest or thrown straight up or down. There are now two dimensions to the motion which act independently of each other - that is, neither the projectile's horizontal velocity nor its horizontal position impact its instantaneous vertical velocity or position.
Before we work any mathematical problems involving projectiles released at an angle, let's first review the graphical behavior of objects exhibiting a uniform, negative acceleration and objects moving at a uniform speed.
A Graphical Review of Vertical Properties
Vertically, gravity will still accelerate the projectile at -9.8 m/sec2.
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losing speed while traveling in a positive direction
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position-time
s-t
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velocity-time
v-t
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acceleration-time
a-t
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gaining speed while traveling in a negative direction
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position-time
s-t
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velocity-time
v-t
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acceleration-time
a-t
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A Graphical Review of Horizontal Properties
But, horizontally, there is no acceleration since gravity acts at right angles to that velocity's component and there is no air resistance. Note in the original diagram shown at the top of the page that the horizontal spacing is uniform, that is, the projectile travels the same distance forward each second.
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traveling at a constant speed in a positive direction
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position-time
s-t
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velocity-time
v-t
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acceleration-time
a-t
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Projectiles Released at an Angle
Consequently, when a projectile is released at an angle with a given speed its trajectory must be analyzed in two parts. Play the following
physlet from Davidson College showing a ball thrown at an angle from the ground and note how these two simultaneous, yet independent, behaviors work together.
In solving these problems we can always start with the following conditions:
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Horizontal
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Vertical
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time
release speed
release angle
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| a = 0 |
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a = - 9.8 m/sec2 |
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vH = v cos(θ)
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vv = vo = v sin(θ)
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R = vHt
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s = vot + ½at2 |
In the table shown above, the variable, R, represents the range of the projectile; that is, the horizontal distance that the projectile travels from the point of release until it strikes the ground. Since there is no horizontal acceleration, the formula used to calculate a projectile's range,
R = vHt
, is derived from the equation d = rt for constant speed in which we substitute in range for distance, vH for rate, and leave time the same.
Notice, that the horizontal component vH is not renamed since it remains constant throughout the projectile's entire trajectory while vv is renamed as vo since the vertical velocity is constantly changing and vv is just the vertical component of its initial speed as it begins its trajectory.
Note that the time, t, the initial release speed, v, and the trajectory's angle, θ, cross between the columns since they are parameters that apply to the bahaviors in both columns; that is, they are common quantities.
You might want to take a moment to review your
kinematics equations for uniformly accelerated motion prior to completing the next examples. To solve these problems we will use an
H | V chart modeling the one shown above. This type of chart provides us a means of organizing our information and helps us apply the correct formulas for each component's behavior.