Acceleration in Projectile Motion
Acceleration in projectile motion refers to the rate of change of velocity as an object moves through the air. In the vertical direction, the acceleration is due to gravity, causing the object to accelerate downward at a constant rate. In the horizontal direction, there is no acceleration (assuming no air resistance), so the velocity remains constant.
6 Key excerpts on "Acceleration in Projectile Motion"
eBook - ePubInstant Notes in Sport and Exercise Biomechanics
Second Edition
- Paul Grimshaw, Michael Cole, Adrian Burden, Neil Fowler(Authors)
- 2019(Publication Date)
- Garland Science(Publisher)
...At higher velocities, that are more realistic for shot-putters (e.g. 10–15 m/s), the same changes in velocity or release height have a much smaller influence on the optimal angle of release (Figure A6.7). Key notes Introduction Newton’s Second Law of Motion dictates that bodies that experience a constant force also accelerate at a constant rate. The most common example of this occurring on Earth is when a body is airbourne, where the attractive force between the body and the Earth provides an acceleration equal to –9.81 m/s 2. Effects of constant acceleration A body subjected to a constant acceleration will experience a linear change in velocity and a curvilinear change in position when viewed over time. At any point in time the motion of a body (e.g. its position or velocity) that is accelerating constantly can be calculated using one of Galileo’s equations of uniformly accelerated motion. These equations can be used to find, for example, the height raised by an athlete’s centre of mass (COM) during a jump. Projectile motion A projectile is a body that is unsupported (e.g. a ball in flight) and is only affected by the forces associated with gravity and air resistance. Projectiles generally have both horizontal and vertical velocity components during flight. If air resistance can be ignored, the horizontal velocity remains constant and the vertical velocity is affected by the constant acceleration due to gravity, which results in the projectile having a parabolic flight path. Maximising the range of a projectile For a body that lands at the same height that it was projected from, its range is dependent upon both its velocity and angle of projection at take-off. More specifically, range is proportional to the square of the take-off velocity, so higher velocities will result in proportionally greater gains in range...
eBook - ePub- William Bolton(Author)
- 2012(Publication Date)
- Routledge(Publisher)
...The horizontal component of the velocity is u cos θ and the vertical component is u sin θ. There is no gravitational force acting in the horizontal direction and so there is no acceleration imposed on the motion in that direction. The horizontal velocity thus remains, when we neglect air resistance, constant. The vertical component is, however, subject to gravity and so there is a constant acceleration. Figure 4.16 Projectile For a projectile: the horizontal motion is with constant velocity but the vertical motion is one of uniform acceleration. Thus, for a projectile after a time t from its projection, the horizontal distance x from the start is: x = u cos θ x t [12] There is a vertical force of gravity and so the motion in the vertical direction suffers an acceleration. Thus after a time t, equation [ 9 ], i.e. s = ut + ½ at 2, gives for the vertical distance y travelled: y = u sin θ × t − ½ gt 2 [13] where g is the acceleration due to gravity. The maximum height h reached by the projectile is when the vertical component of the velocity is zero. Thus, using equation [ 11 ], i.e. v 2 = u 2 + 2 as, gives: 0 = (u sin θ) 2 − 2 gh Hence: [14] At the end of the flight the vertical displacement is zero. Thus, if T is the time of flight, equation [ 9 ], i.e. s = ut + ½ at 2, gives: 0 = u sin θ × T − ½ gT 2 Hence: [15] The horizontal distance, i.e. the range, R covered in time T is the horizontal velocity multiplied by the time and thus: Since 2 cos θ sin θ = sin 2θ, then: [16] Application There are many situations where projectiles might be required to have their maximum range. For example, a shot putter at an athletics meeting wants to project the shot as far as possible. He/she has control over the speed of projection and the angle to the horizontal at which to project the shot. If we consider the problem as a projectile being projected from ground level then the maximum horizontal distance will be achieved with a launch angle of 45°...
eBook - ePub- Richard Gentle, Peter Edwards, William Bolton(Authors)
- 2001(Publication Date)
- Newnes(Publisher)
...If an object is allowed to fall in air then it accelerates vertically downwards at a rate of approximately 9.81 m/s 2. After falling for quite a time the velocity can build up to such a point that the resistance from the air becomes large and the acceleration decreases. Eventually the object can reach what is called a terminal velocity and there is no further acceleration. An example of this is a free-fall parachutist. For most examples of interest to engineers, however, the acceleration due to gravity can be taken as constant and continuous. They are worthwhile considering, therefore, as a separate case. The first thing to note is that the acceleration due to gravity is the same for any object, which is why heavy objects only fall at the same rate as light ones. In practice a feather will not fall as fast as a football, but that is simply because of the large air resistance associated with a feather which produces a very low terminal velocity. Therefore we generally do not need to consider the mass of the object. Problems involving falling masses therefore do not involve forces and are examples of kinematics which can be analysed by the equations of uniform motion developed above. This does not necessarily restrict us to straight line movement, as we saw in the worked example, and so we can look briefly at the subject of trajectories. Example 4.1.2 Suppose a tennis ball is struck so that it leaves the racquet horizontally with a velocity of 25 m/s and at a height above the ground of 1.5 m. How far will the ball travel horizontally before it hits the ground? Clearly the motion of the ball will be a curve, known as a trajectory, because of the influence of gravity. At first sight this does not seem like the sort of problem that can be analysed by the equations of uniform motion in a straight line. However, the ball’s motion can be resolved into horizontal motion and vertical motion, thus allowing direct application of the four equations that are at our disposal...
eBook - ePub- Alan Hendrickson(Author)
- 2012(Publication Date)
- Routledge(Publisher)
...The rate at which velocity changes with time is important too, and this is called acceleration. Average acceleration is defined as: a ¯ = Δ v Δ t Where a ¯ = average acceleration (ft/sec 2, m/sec 2) v = velocity (ft/sec, m/sec) t = time (seconds) Using the same logic as was used earlier in the discussion on velocity, because the finite time interval Δ t is involved, this is an average acceleration. Instantaneous acceleration will be obtained only if Δ t is infinitesimally larger than zero. a = l i m Δ t → 0 Δ v Δ t = d v d t If there is no change in velocity, meaning that velocity is constant, or stated mathematically Δ v = 0, then a = 0. Acceleration is the ratio of velocity change to time. The units of acceleration are therefore the units of velocity divided by the unit of time. This is expressed one of two ways, displacement per time per time, or displacement per time squared. Commonly this would be stated as “feet per second per second” or “feet per second squared”, and written as ft/sec/sec or ft/sec 2 (or likewise m/sec/sec, or m/sec 2). The rate at which acceleration changes with respect to time is somewhat amusingly called jerk. This quantity is rarely used but deserves mention because its presence is very perceptible. Jerky motion can result from elastic or springy elements within a mechanical system. Any blockage of movement is temporarily allowed by the springy element, but eventually forces build up and overcome the blockage and the springs stored force is released, causing the load to surge ahead...
eBook - ePub- W. Bolton(Author)
- 2015(Publication Date)
- Routledge(Publisher)
...A vector quantity is one for which both its magnitude and direction have to be stated for its effects to be determined; they have to be added by methods which take account of their directions, e.g. the parallelogram method. Figure 4.1 Example Example A projectile is thrown vertically upwards with a velocity of 10 m/s. If there is a horizontal wind blowing at 5 m/s, what will be the velocity with which the projectile starts out? Figure 4.1 shows the vectors representing the two velocities and their resultant, i.e. sum, determined from the parallelogram of vectors. We can use a scale drawing to obtain the resultant or, because the angle between the two velocities is 90º, we can use the Pythagoras theorem to give v 2 = 10 2 + 5 2. Hence v = 11.2 m/s. This velocity will be at an angle θ to the horizontal, where tan θ = 10/5 and so θ 5 63.4º. 4.3.1 Resolution into components A single vector can be resolved into two components at right angles to each other by using the parallelogram of vector method of summing vectors in reverse. For example, a velocity v at an angle θ to the horizontal can be resolved into a horizontal component of v cos θ and a vertical component of v sin θ (Figure 4.2). Figure 4.2 Resolving a velocity into components Example A projectile is fired from a gun with a velocity of 200 m/s at 30º to the horizontal. What is (a) the horizontal velocity component, (b) the vertical velocity component? Horizontal component 5 v cos θ = 200 cos 30º = 173 m/s. Vertical component 5 v sin θ = 200 sin 30º = 100 m/s. 4.4 Motion under gravity All freely falling objects in a vacuum fall with the same uniform acceleration directed towards the surface of the earth as a result of a gravitational force acting between the object and the earth. This acceleration is termed the acceleration due to gravity g. For most practical purposes, the acceleration due to gravity at the surface of the earth is taken as being 9.81 m/s 2...
eBook - ePub- A. D. Johnson(Author)
- 2017(Publication Date)
- CRC Press(Publisher)
...Motion of bodies 2 2.1 Aims To introduce the concepts of linear and angular motion. To explain the relationships between displacement, velocity and acceleration. To explain the relationship between absolute and relative velocities. To define the equations used to analyse linear and angular motion. To introduce an approach by which linear and angular motion problems can be analysed. To explain related topics such as ‘falling bodies’, ‘trajectories’ and vector methods. 2.2 Introduction to Motion When traffic lights turn to green a car will move away with increasing velocity. The car will cover a distance in a particular direction and will possess a particular velocity at any instant. During this process the car possesses the three basic constituents of motion, namely: displacement, velocity and acceleration. It should be noted that since the car runs on wheels, these will also be in motion and therefore possess displacement, velocity and acceleration. However, the car moves in a linear direction, while the wheels move in an angular direction. 2.2.1 Displacement If a man walks 10 km, there is an indication of the distance between the start position and the final position, but there is no indication of the direction. The 10 km is merely the distance covered and, as such, is a scalar quantity, i.e. possessing magnitude only. Displacement, however, implies a change in position or movement over a distance and gives the position and direction from the start point. Thus displacement is a vector quantity possessing both magnitude and direction. Fig. 2.1 Displacement diagram. Figure 2.1 gives an example of a man who walks 3 km east then 4 km north. He has actually walked a distance of 7 km but has been displaced from his start point by only 5 km. 2.2.2 Velocity Velocity is the value of displacement measured over a period of time. It is the rate over which a distance/displacement is traversed...
