An object that is thrown vertically upward is also a projectile provided that the influence of air resistance is negligible. The applications of projectile motion in physics and engineering are numerous. Such objects are called projectiles and their path is called a trajectory. For example, when you jump, your legs apply a force to the ground, and the ground applies and equal and opposite reaction force that propels you into the air.
A projectile is an object upon which the only force is gravity. This fact was discussed in Kinematics in Two Dimensions: An Introduction , where vertical and horizontal motions were seen to be independent. 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. This choice of axes is the most sensible, because acceleration due to gravity is vertical—thus, there will be 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. Figure 1 illustrates the notation for displacement, where s is defined to be the total displacement and x and y are its components along the horizontal and vertical axes, respectively. The magnitudes of these vectors are s , x , and y. Note that in the last section we used the notation A to represent a vector with components A x and A y.
If we continued this format, we would call displacement s with components s x and s y. However, to simplify the notation, we will simply represent the component vectors as x and y.
Of course, to describe motion we must deal with velocity and acceleration, as well as with displacement. We must find their components along the x — and y -axes, too. We will assume all forces except gravity such as air resistance and friction, for example are negligible. Note that this definition assumes that the upwards direction is defined as the positive direction.
If you arrange the coordinate system instead such that the downwards direction is positive, then acceleration due to gravity takes a positive value. Both accelerations are constant, so the kinematic equations can be used. Figure 1. The total displacement s of a soccer ball at a point along its path. The vector s has components x and y along the horizontal and vertical axes. Step 1. Resolve or break the motion into horizontal and vertical components along the x- and y-axes.
The magnitude of the components of displacement s along these axes are x and y. Initial values are denoted with a subscript 0, as usual. Step 2. Treat the motion as two independent one-dimensional motions, one horizontal and the other vertical. The kinematic equations for horizontal and vertical motion take the following forms:. Step 3. Solve for the unknowns in the two separate motions—one horizontal and one vertical. Note that the only common variable between the motions is time t. The problem solving procedures here are the same as for one-dimensional kinematics and are illustrated in the solved examples below.
Step 4. Recombine the two motions to find the total displacement s and velocity v. Figure 2. As the object falls towards the Earth again, the vertical velocity increases again in magnitude but points in the opposite direction to the initial vertical velocity. During a fireworks display, a shell is shot into the air with an initial speed of The fuse is timed to ignite the shell just as it reaches its highest point above the ground.
Because air resistance is negligible for the unexploded shell, the analysis method outlined above can be used. We can then define x 0 and y 0 to be zero and solve for the desired quantities. Since we know the initial and final velocities as well as the initial position, we use the following equation to find y :. Figure 3. The trajectory of a fireworks shell. The fuse is set to explode the shell at the highest point in its trajectory, which is found to be at a height of m and m away horizontally.
Because y 0 and v y are both zero, the equation simplifies to. Now we must find v 0 y , the component of the initial velocity in the y -direction. Note that because up is positive, the initial velocity is positive, as is the maximum height, but the acceleration due to gravity is negative. Note also that the maximum height depends only on the vertical component of the initial velocity, so that any projectile with a The numbers in this example are reasonable for large fireworks displays, the shells of which do reach such heights before exploding.
In practice, air resistance is not completely negligible, and so the initial velocity would have to be somewhat larger than that given to reach the same height. As in many physics problems, there is more than one way to solve for the time to the highest point.
Because y 0 is zero, this equation reduces to simply. Note that the final vertical velocity, v y , at the highest point is zero. This time is also reasonable for large fireworks. When you are able to see the launch of fireworks, you will notice several seconds pass before the shell explodes. The horizontal motion is a constant velocity in the absence of air resistance. The horizontal displacement found here could be useful in keeping the fireworks fragments from falling on spectators.
Once the shell explodes, air resistance has a major effect, and many fragments will land directly below. If our thought experiment continues and we project the cannonball horizontally in the presence of gravity, then the cannonball would maintain the same horizontal motion as before - a constant horizontal velocity. Furthermore, the force of gravity will act upon the cannonball to cause the same vertical motion as before - a downward acceleration. The cannonball falls the same amount of distance as it did when it was merely dropped from rest refer to diagram below.
However, the presence of gravity does not affect the horizontal motion of the projectile. The force of gravity acts downward and is unable to alter the horizontal motion. There must be a horizontal force to cause a horizontal acceleration.
And we know that there is only a vertical force acting upon projectiles. The vertical force acts perpendicular to the horizontal motion and will not affect it since perpendicular components of motion are independent of each other. Thus, the projectile travels with a constant horizontal velocity and a downward vertical acceleration. Now suppose that our cannon is aimed upward and shot at an angle to the horizontal from the same cliff. In the absence of gravity i. An object in motion would continue in motion at a constant speed in the same direction if there is no unbalanced force.
This is the case for an object moving through space in the absence of gravity. However, if the gravity switch could be turned on such that the cannonball is truly a projectile, then the object would once more free-fall below this straight-line, inertial path. In fact, the projectile would travel with a parabolic trajectory. The downward force of gravity would act upon the cannonball to cause the same vertical motion as before - a downward acceleration.
If there were any other force acting upon an object, then that object would not be a projectile. Thus, the free-body diagram of a projectile would show a single force acting downwards and labeled force of gravity or simply F grav. Regardless of whether a projectile is moving downwards, upwards, upwards and rightwards, or downwards and leftwards, the free-body diagram of the projectile is still as depicted in the diagram at the right.
By definition, a projectile is any object upon which the only force is gravity. Many students have difficulty with the concept that the only force acting upon an upward moving projectile is gravity. Their conception of motion prompts them to think that if an object is moving upward, then there must be an upward force. And if an object is moving upward and rightward, there must be both an upward and rightward force.
Their belief is that forces cause motion; and if there is an upward motion then there must be an upward force. They reason, "How in the world can an object be moving upward if the only force acting upon it is gravity? Newton's laws suggest that forces are only required to cause an acceleration not a motion. Recall from the Unit 2 that Newton's laws stood in direct opposition to the common misconception that a force is required to keep an object in motion.
This idea is simply not true! A force is not required to keep an object in motion. A force is only required to maintain an acceleration. And in the case of a projectile that is moving upward, there is a downward force and a downward acceleration.
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