The initial velocity can be broken down using an equation relating the sine and cosine:
The velocity that is given has both x and y components, because it is in a direction 60.0° up from the horizontal (x) direction. How much time does the toy rocket spend the air, and how far from its launch point does it land in the field?Īnswer: The first thing that must be found to solve this problem is the initial velocity in the x and y directions. Its initial velocity has a magnitude of 20.0 m/s. Therefore, the velocity (magnitude and direction) of the ball after 5.00 s was 51.24 m/s, -72.98° down from the horizontal.Ģ) A toy rocket is launched in a flat field, aimed at an angle 60.0° up from the horizontal (x) axis. If the horizontal direction is 0.0 radians, the angle can be found with the equation: Though it was not asked for in the question, it is also possible to find the direction of the velocity as an angle. The magnitude of the velocity is 51.24 m/s. To find the magnitude of the velocity, the x and y components must be added with vector addition: In projectile motion problems, up is defined as the positive direction, so the y component has a magnitude of 49.0 m/s, in the down direction. The x component of the velocity after 5.00 s is: The ball was kicked horizontally, so v xo = 15.0 m/s, and v yo = 0.0 m/s.
Once these two components are found, they must be combined using vector addition to find the final velocity. After 5.00 s, what is the magnitude of the velocity of the ball?Īnswer: The velocity of the ball after 5.00 s has two components. The initial velocity of the ball is 15.0 m/s horizontally. G = acceleration due to gravity (9.80 m/s 2)ġ) A child kicks a soccer ball off of the top of a hill. Vertical velocity = initial vertical velocity - (acceleration due to gravity)(time) Horizontal velocity = initial horizontal velocity Horizontal distance = (initial horizontal velocity)(time) The horizontal and vertical velocities are expressed in meters per second (m/s). The units to express the horizontal and vertical distances are meters (m). Velocity is a vector (it has magnitude and direction), so the overall velocity of an object can be found with vector addition of the x and y components: v 2 = v x 2 + v y 2. The trajectory has horizontal (x) and vertical (y) components. The path the object follows is determined by these effects (ignoring air resistance). In other cases we may choose a different set of axes.A projectile is an object that is given an initial velocity, and is acted on by gravity. It is not required that we use this choice of axes it is simply convenient in the case of gravitational acceleration. (This choice of axes is the most sensible because acceleration resulting from gravity is vertical thus, there is no acceleration along the horizontal axis when air resistance is negligible.) As is customary, we call the horizontal axis the x-axis and the vertical axis the y-axis. The key to analyzing two-dimensional projectile motion is to break it into two motions: one along the horizontal axis and the other along the vertical.
We discussed this fact in Displacement and Velocity Vectors, where we saw that vertical and horizontal motions are independent. The most important fact to remember here is that motions along perpendicular axes are independent and thus can be analyzed separately. In this section, we consider two-dimensional projectile motion, and our treatment neglects the effects of air resistance. The motion of falling objects as discussed in Motion Along a Straight Line is a simple one-dimensional type of projectile motion in which there is no horizontal movement. Such objects are called projectiles and their path is called a trajectory. Some examples include meteors as they enter Earth’s atmosphere, fireworks, and the motion of any ball in sports. The applications of projectile motion in physics and engineering are numerous. Projectile motion is the motion of an object thrown or projected into the air, subject only to acceleration as a result of gravity. Calculate the trajectory of a projectile.Find the time of flight and impact velocity of a projectile that lands at a different height from that of launch.Calculate the range, time of flight, and maximum height of a projectile that is launched and impacts a flat, horizontal surface.Use one-dimensional motion in perpendicular directions to analyze projectile motion.By the end of this section, you will be able to:
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