Time at which projectile hits the ground

Table of contents: Time of flight equation Time of flight exemplary calculations. Time of flight equation To define the time of flight equation, we should split the formulas into two cases: 1. Time of flight exemplary calculations Let's use this time of flight calculator to find out how long it takes for a pebble thrown from the edge of the Grand Canyon to hit the ground.

Type in the velocity value. Enter the angle. Finally, type in the initial height. Let's take the deepest point of the Canyon. It's 6, ft difference - over a mile! When the point of projection and point of return are on the same horizontal plane, the net vertical displacement of the object is zero. All projectile motion happens in a bilaterally symmetrical path, as long as the point of projection and return occur along the same horizontal surface. Bilateral symmetry means that the motion is symmetrical in the vertical plane.

If you were to draw a straight vertical line from the maximum height of the trajectory, it would mirror itself along this line. As the projectile moves upwards it goes against gravity, and therefore the velocity begins to decelerate. Eventually the vertical velocity will reach zero, and the projectile is accelerated downward under gravity immediately.

Once the projectile reaches its maximum height, it begins to accelerate downward. This is also the point where you would draw a vertical line of symmetry. The range of the projectile is the displacement in the horizontal direction. There is no acceleration in this direction since gravity only acts vertically. Like time of flight and maximum height, the range of the projectile is a function of initial speed. Range : The range of a projectile motion, as seen in this image, is independent of the forces of gravity.

Privacy Policy. Skip to main content. Two-Dimensional Kinematics. Search for:. Projectile Motion. Basic Equations and Parabolic Path Projectile motion is a form of motion where an object moves in parabolic path; the path that the object follows is called its trajectory. Learning Objectives Assess the effect of angle and velocity on the trajectory of the projectile; derive maximum height using displacement.

Key Takeaways Key Points Objects that are projected from, and land on the same horizontal surface will have a vertically symmetrical path. The time it takes from an object to be projected and land is called the time of flight. This depends on the initial velocity of the projectile and the angle of projection.

When the projectile reaches a vertical velocity of zero, this is the maximum height of the projectile and then gravity will take over and accelerate the object downward. The horizontal displacement of the projectile is called the range of the projectile, and depends on the initial velocity of the object.

Key Terms trajectory : The path of a body as it travels through space. Solving Problems In projectile motion, an object moves in parabolic path; the path the object follows is called its trajectory. Learning Objectives Identify which components are essential in determining projectile motion of an object.

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. It is not required that we use this choice of axes; it is simply convenient in the case of gravitational acceleration. In other cases we may choose a different set of axes.

Figure illustrates the notation for displacement, where we define. The magnitudes of these vectors are s , x , and y. To describe projectile motion completely, we must include velocity and acceleration, as well as displacement.

We must find their components along the x- and y -axes. Defining the positive direction to be upward, the components of acceleration are then very simple:. With these conditions on acceleration and velocity, we can write the kinematic Equation through Equation for motion in a uniform gravitational field, including the rest of the kinematic equations for a constant acceleration from Motion with Constant Acceleration. The kinematic equations for motion in a uniform gravitational field become kinematic equations with.

Using this set of equations, we can analyze projectile motion, keeping in mind some important points. Treat the motion as two independent one-dimensional motions: one horizontal and the other vertical. Use the kinematic equations for horizontal and vertical motion presented earlier. 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 those for one-dimensional kinematics and are illustrated in the following solved examples. Recombine quantities in the horizontal and vertical directions to find the total displacement.

At its highest point, the vertical velocity is zero. As the object falls toward Earth again, the vertical velocity increases again in magnitude but points in the opposite direction to the initial vertical velocity. Example A Fireworks Projectile Explodes High and Away 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.

The highest point in any trajectory, called the apex , is reached when. Since we know the initial and final velocities, as well as the initial position, we use the following equation to find y :. Note that because up is positive, the initial vertical velocity is positive, as is the maximum height, but the acceleration resulting from 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, so the initial velocity would have to be somewhat larger than that given to reach the same height.

In this case, the easiest method is to use. This time is also reasonable for large fireworks. If you are able to see the launch of fireworks, notice that several seconds pass before the shell explodes. Another way of finding the time is by using. The horizontal displacement is the horizontal velocity multiplied by time as given by. 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. When the shell explodes, air resistance has a major effect, and many fragments land directly below.

Note that the angle for the displacement vector is less than the initial angle of launch. To see why this is, review Figure , which shows the curvature of the trajectory toward the ground level. When solving Figure a , the expression we found for y is valid for any projectile motion when air resistance is negligible. This equation defines the maximum height of a projectile above its launch position and it depends only on the vertical component of the initial velocity.

On its way down, the ball is caught by a spectator 10 m above the point where the ball was hit. Again, resolving this two-dimensional motion into two independent one-dimensional motions allows us to solve for the desired quantities. The time a projectile is in the air is governed by its vertical motion alone.

Thus, we solve for t first. While the ball is rising and falling vertically, the horizontal motion continues at a constant velocity. This example asks for the final velocity. Thus, we recombine the vertical and horizontal results to obtain. We can find the time for this by using Figure :. The initial vertical velocity is the vertical component of the initial velocity:. Substituting into Figure for y gives us. Since the ball is at a height of 10 m at two times during its trajectory—once on the way up and once on the way down—we take the longer solution for the time it takes the ball to reach the spectator:.

The time for projectile motion is determined completely by the vertical motion. Thus, any projectile that has an initial vertical velocity of The service line is Will the ball land in the service box, whose out line is 6. A football quarterback is moving straight backward at a speed of 2. Gun sights are adjusted to aim high to compensate for the effect of gravity, effectively making the gun accurate only for a specific range.

An eagle is flying horizontally at a speed of 3. Calculate the velocity of the fish relative to the water when it hits the water. An owl is carrying a mouse to the chicks in its nest. Its position at that time is 4. The owl is flying east at 3. Is the owl lucky enough to have the mouse hit the nest? To answer this question, calculate the horizontal position of the mouse when it has fallen Suppose a soccer player kicks the ball from a distance 30 m toward the goal.

Find the initial speed of the ball if it just passes over the goal, 2. The distance will be about 95 m. The free throw line in basketball is 4. A player standing on the free throw line throws the ball with an initial speed of 7. At what angle above the horizontal must the ball be thrown to exactly hit the basket? Note that most players will use a large initial angle rather than a flat shot because it allows for a larger margin of error.

Explicitly show how you follow the steps involved in solving projectile motion problems. In , Michael Carter U. What was the initial speed of the shot if he released it at a height of 2. A basketball player is running at 5.

He maintains his horizontal velocity. Without an effect from the wind, the ball would travel What distance does the ball travel horizontally? These equations describe the x and y positions of a projectile that starts at the origin. Unreasonable Results a Find the maximum range of a super cannon that has a muzzle velocity of 4. Explain your answer. Construct Your Own Problem Consider a ball tossed over a fence.

Among the things to determine are; the height of the fence, the distance to the fence from the point of release of the ball, and the height at which the ball is released. You should also consider whether it is possible to choose the initial speed for the ball and just calculate the angle at which it is thrown. Also examine the possibility of multiple solutions given the distances and heights you have chosen. Air resistance would have the effect of decreasing the time of flight, therefore increasing the vertical deviation.

Skip to main content. Two-Dimensional Kinematics. Search for:. Projectile Motion Learning Objectives By the end of this section, you will be able to: Identify and explain the properties of a projectile, such as acceleration due to gravity, range, maximum height, and trajectory.

Determine the location and velocity of a projectile at different points in its trajectory. Apply the principle of independence of motion to solve projectile motion problems. Example 1. A Fireworks Projectile Explodes High and Away During a fireworks display, a shell is shot into the air with an initial speed of Strategy Because air resistance is negligible for the unexploded shell, the analysis method outlined above can be used.

Since we know the initial and final velocities as well as the initial position, we use the following equation to find y : Figure 3. Defining a Coordinate System It is important to set up a coordinate system when analyzing projectile motion. One part of defining the coordinate system is to define an origin for the x and y positions. It is also important to define the positive and negative directions in the x and y directions.

When this is the case, the vertical acceleration, g , takes a negative value since it is directed downwards towards the Earth. However, it is occasionally useful to define the coordinates differently. For example, if you are analyzing the motion of a ball thrown downwards from the top of a cliff, it may make sense to define the positive direction downwards since the motion of the ball is solely in the downwards direction.

If this is the case, g takes a positive value. Example 2. Figure 4. The trajectory of a rock ejected from the Kilauea volcano. One of the most important things illustrated by projectile motion is that vertical and horizontal motions are independent of each other. Galileo was the first person to fully comprehend this characteristic.