The pitcher in a slow-pitch softball game releases the ball at a point above ground level. A stroboscopic plot of the position of the ball is shown in Fig. 4-60, where the readings are apart and the ball is released at. (a) What is the initial speed of the ball? (b) What is the speed of the ball at the instant it reaches its maximum height above ground level? (c) What is that maximum height?
Kinematic equations are the equations of motion that relate initial velocity, final velocity, acceleration, displacement, and time.
We can find the initial speed of the ball using the kinematic equation of distance. Using time and respective distance we can find the speed at maximum height and maximum height.
The second kinematic equation of motion, (i)
Here, is final displacement, is initial displacement, is initial velocity, is time and is acceleration.
From the graph, we can find the total time is and the horizontal distance is.
We can find the horizontal component of initial velocity, along the horizontal direction using equation (i). Substitute the given values in equation (i).
Similarly, we can find the initial vertical velocity component, ball come at the same level by traveling vertical distance using equation (i). Therefore, substitute the given values in equation (i) to calculate the initial vertical velocity.
Hence, the magnitude of the initial speed is calculated by substituting the value of vertical and horizontal velocity in the equation for the magnitude.
Hence, the value of initial speed is .
At maximum height vertical velocity becomes and there is only horizontal velocity component so the speed at the maximum height is .
Here we can use the vertical velocity component and half time of flight to find maximum height. Substituting the values in equation (i), we get
Hence, the value of the maximum height of the ball is 9.3 ft.
The acceleration of a particle moving only on a horizontal xy plane is given by , where is in meters per second squared and t is in seconds. At , the position vector locates the particle, which then has the velocity vector . At , what are (a) its position vector in unit-vector notation and (b) the angle between its direction of travel and the positive direction of the x axis.
A trebuchet was a hurling machine built to attack the walls of a castle under siege. A large stone could be hurled against a wall to break apart the wall. The machine was not placed near the wall because then arrows could reach it from the castle wall. Instead, it was positioned so that the stone hit the wall during the second half of its flight. Suppose a stone is launched with a speed of and at an angle of. What is the speed of the stone if it hits the wall (a) just as it reaches the top of its parabolic path and (b) when it has descended to half that height? (c)As a percentage, how much faster is it moving in part (b) than in part (a)?
Suppose that a shot putter can put a shot at the world-class speed and at a height of. What horizontal distance would the shot travel if the launch angleis (a)and(b)? The answers indicate that the angle of, which maximizes the range of projectile motion, does not maximize the horizontal distance when the launch and landing are at different heights.
A helicopter is flying in a straight line over a level field at a constant speed of and at a constant altitude of . A package is ejected horizontally from the helicopter with an initial velocity of relative to the helicopter and in a direction opposite the helicopter’s motion. (a) Find the initial speed of the package relative to the ground. (b) What is the horizontal distance between the helicopter and the package at the instant the package strikes the ground? (c) What angle does the velocity vector of the package meet with the ground at the instant before impact, as seen from the ground?
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