![]() The data in the table above show the symmetrical nature of a projectile’s trajectory. The following table lists the results of such calculations for the first four seconds of the projectile’s motion. If a projectile is launched with an initial vertical velocity of 19.6 m/s and an initial horizontal velocity of 33.9 m/s, then the x- and y- displacements of the projectile can be calculated using the equations above. Where v iy is the initial vertical velocity in m/s, t is the time in seconds, and g = -9.8 m/s/s (an approximate value of the acceleration of gravity). (equation for vertical displacement for an angled-launched projectile) Combining these two influences upon the vertical displacement yields the following equation. In the presence of gravity, it will fall a distance of 0.5 ![]() In the absence of gravity, a projectile would rise a vertical distance equivalent to the time multiplied by the vertical component of the initial velocity (v iy ![]() However, the gravity-free path is no longer a horizontal line since the projectile is not launched horizontally. The projectile still falls below its gravity-free path by a vertical distance of 0.5*g*t^2. These distances are indicated on the diagram below. The projectile still falls 4.9 m, 19.6 m, 44.1 m, and 78.4 m below the straight-line, gravity-free path. How will the presence of an initial vertical component of velocity affect the values for the displacement? The diagram below depicts the position of a projectile launched at an angle to the horizontal. Now consider displacement values for a projectile launched at an angle to the horizontal (i.e., a non-horizontally launched projectile). This information is summarized in the table below. Thus, the horizontal displacement is 20 m at 1 second, 40 meters at 2 seconds, 60 meters at 3 seconds, etc. This is consistent with the initial horizontal velocity of 20 m/s. Furthermore, since there is no horizontal acceleration, the horizontal distance traveled by the projectile each second is a constant value – the projectile travels a horizontal distance of 20 meters each second. It can also be seen that the vertical displacement follows the equation above (y = 0.5 In this example, the initial horizontal velocity is 20 m/s and there is no initial vertical velocity (i.e., a case of a horizontally launched projectile).Īs can be seen in the diagram above, the vertical distance fallen from rest during each consecutive second is increasing (i.e., there is a vertical acceleration). The position of the object at 1-second intervals is shown. t The diagram below shows the trajectory of a projectile (in red), the path of a projectile released from rest with no horizontal velocity (in blue) and the path of the same object when gravity is turned off (in green).Thus, if the horizontal displacement ( x) of a projectile were represented by an equation, then that equation would be written as The horizontal displacement of a projectile is only influenced by the speed at which it moves horizontally ( v ix) and the amount of time ( t) that it has been moving horizontally. It was also discussed earlier, that the force of gravity does not influence the horizontal motion of a projectile. The above equation pertains to a projectile with no initial vertical velocity and as such predicts the vertical distance that a projectile falls if dropped from rest. t 2 (equation for vertical displacement for a horizontally launched projectile)where g is -9.8 m/s/s and t is the time in seconds.This equation was discussed in Unit 1 of The Physics Classroom. Thus, the vertical displacement ( y) of a projectile can be predicted using the same equation used to find the displacement of a free-falling object undergoing one-dimensional motion. ![]() As has already been discussed, the vertical displacement (denoted by the symbol y in the discussion below) of a projectile is dependent only upon the acceleration of gravity and not dependent upon the horizontal velocity. Now we will investigate the manner in which the horizontal and vertical components of a projectile’s displacement vary with time. How the horizontal and vertical components of the velocity vector change with time during the course of projectile’s trajectory. ![]()
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