A ball player hits a home run, and the baseball just clears a wall 17.3 m high located 131.0 m from home plate. The ball is hit an an angle of 35° to the horizontal, and air resistance is negligible. Assume the ball is hit at a height of 1.0 m above the ground.
(a) What is the initial speed?
m/s
(b) How much time does it take for the ball to reach the wall?
s
(c) Find the components of the velocity and the speed of the ball when it reaches the wall
vy,f = m/s
vx,f = m/s
vf = m/s
5 m/s, 8.5 m/s
To solve this problem, we can use the equations of motion in projectile motion. Since air resistance is negligible, we can assume that the horizontal and vertical motions are independent.
(a) To find the initial speed (Vo), we can use the horizontal velocity component:
Vx = Vo * cos(θ)
where Vx is the horizontal velocity component and θ is the angle of elevation.
Given values:
θ = 35°
Vx = ?
Rearranging the equation, we can solve for Vx:
Vx = V / cos(θ)
It's important to note that the horizontal velocity remains constant throughout the projectile motion, so we can use the value of Vx at any point in the motion. In this case, since it is hitting the wall, we can use the horizontal velocity at that point:
Vx = Vf (final horizontal velocity)
(b) To find the time of flight (t), we can use the vertical velocity component:
Vy = Vo * sin(θ)
Using the equation:
y = y0 + Vyo * t - (1/2) * g * t^2
where
y = displacement in the y-direction
y0 = initial vertical position
Vyo = initial vertical velocity component
g = acceleration due to gravity (-9.8 m/s^2)
t = time
Given values:
y0 = 1.0 m
Vy = Vf * sin(θ)
g = -9.8 m/s^2
t = ?
Simplifying the equation for y = 0 (when the ball reaches the wall):
0 = 1.0 + Vy * t - 4.9 * t^2
This is a quadratic equation that can be solved to find the time of flight, t.
(c) To find the final velocity components at the wall, we can use the following equations:
Vy,f = Vy - g * t
Vx,f = Vx
Vy,f is the final vertical velocity component, and Vx,f is the final horizontal velocity component.
The magnitude of the final velocity (Vf) can be found using the formula:
Vf = sqrt(Vx,f^2 + Vy,f^2)
Solving these equations will give us the final answers.
Let's calculate the values.
(a) Initial speed (Vo):
Vx = Vf (final horizontal velocity)
Vx = Vo * cos(θ)
Since Vo is the initial speed, we can solve for it:
Vo = Vx / cos(θ)
(b) Time of flight (t):
Using the quadratic equation:
0 = 1.0 + Vy * t - 4.9 * t^2
Solve for t using the quadratic formula.
(c) Final velocity components:
Vy,f = Vy - g * t
Vx,f = Vx
Vf = sqrt(Vx,f^2 + Vy,f^2)