![]() From this information the components of thevelocity at time t = 0 s can be calculated: Note: Often, the total velocity v 0 of the object attime t = 0 s and the angle between the direction of the projectile andthe positive x-axis is provided. We can conclude that the trajectory of the projectile is described by aparabola. ![]() Substituting this expression for t into the equation of motion for y,the following relation between x and y can be obtained: The time t can be eliminated from these two equations: The coordinate system in which we willanalyze the trajectory of the projectile is chosen such that x 0 =y 0 = 0. The trajectory of the projectile is completely determined by theequations of motion x(t) and y(t). In this case, the equations of motion for the projectile are: In describing the motion of the projectile, we will assumethat there is no acceleration in the x-direction, while the acceleration in they-direction is equal to the free-fall acceleration: ![]() Note thata x only affects v x and not v y, anda y only affects v y. Where x 0 and y 0 are the x and y position of theobject at t = 0 s, and v x0 and v y0 are the x and ycomponents of the velocity of the object at time t = 0. Thecoordinate system that will be used to describe the motion of the projectileconsist of an x-axis (horizontal direction) and a y-axis (vertical direction).Assuming that we are dealing with constant acceleration, we can obtain thevelocity and position of the projectile using the procedure outlined in Chapter2: We will start considering the motion of a projectile in 2 dimensions. The velocity of the rabbit has a minimum at t = 14.6 s. To check this prediction, wecalculate the magnitude of the velocity of the rabbit: The scalar product of the velocity and the acceleration will tell ussomething about the change of velocity of the rabbit:įrom this equation we conclude that for t 14.6 s (positive scalarproduct) the rabbits speed will increase. The acceleration of the rabbit is constant (independent of time). The position vector of the rabbit at time t can be expressed as:įrom the equations of motion for x(t) and y(t) we can calculate thevelocity and acceleration: The units of the numerical coefficients in these equations are suchthat, if you substitute t in seconds, x and y are in meters. The path is such that thecomponents of the rabbit's position with respect to the origin of thecoordinate frame are given as function of time by The magnitude of the velocity for small time intervals can be writtenas:Ī rabbit runs across a parking lot on which a set ofcoordinate axes has, strangely enough, been drawn. If we look at a very small time interval, the change in the velocityvector will be small. The magnitude of the velocity as function of time can be calculated: The components of (t)can be calculated from the corresponding components of the position vector (t)and the velocity vector (t):Īssuming a constant acceleration in the x, y and z direction, we canwrite down the following equations of motion: This equation shows that the acceleration of an object in two or threedimensions is also a vector, which can be decomposed into three components: The acceleration of an object inthree dimensions is defined analogously to its definition in Chapter 2: The components of (t)can be calculated from the corresponding components of the position vector (t): Again, the velocity vector canbe decomposed into its three components: This equation shows that the velocity of an object in two or threedimensions is also a vector. The velocity of an object in two or three dimensions is definedanalogously to its definition in Chapter 2: In two or threedimensions, this is much more difficult, and most graphs will show for examplethe trajectory of the object (without providing direct information concerningthe time). ![]() Note: In Chapter 2 we got used to plotting the position of theobject, its velocity and its acceleration as function of time. Using the techniques developed in Chapter 3, we can write the position vector in terms of its components:įigure 4.1. In general, the positionvector will be time dependent (t). The position vector is defined as a vector that starts at the (user defined) origin and ends at thecurrent position of the object (see Figure 4.1). To specify the position of an object the concept of the position vector needs to be introduced. In three dimensions one needs to specifythree coordinates. To answer the question "where am I ?" in two dimensions, oneneeds to specify two coordinates. The discussion about motion in two or three dimensions is morecomplicated. The motion of this object could be described in termsof scalars. Itsposition was unambiguously defined by its distance (positive or negative) froma user defined origin. ![]() In Chapter 2 we discussed the motion of an object in one dimension. ![]()
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