a baseball player hits a home run, and the basketball just clear a wall 22 m loc

Question

a baseball player hits a home run, and the basketball just clear a wall 22 m located 134 m from the home plate. the ball is hit at an angle of 40 degrees to the horizontal, and air resistance is negligible. assume the ball is hit at a height of 1.0 m above the ground. what is the initial speed of the ball? (use g=10 m/s^2) How much time does it take for the ball to reach the wall? Find the components of the velocity and the speed of the ball as it reaches the wall. What is the max height the ball reaches?

Explanation / Answer

Since the air resistance is negligible, the x and y components can be calculated separately.

The ball is hit 35° to the horizontal so the relation between the x and y components of the initial velocity is:

tan 40° = vy / vx

vy = vx * tan 40°

vx is constant, but vy changes with time.

x = 134m

y = (22m - 1m) = 21m

Therefore:

x = vx*t

y = vy*t +1/2*g*t²

Plug in known values:

134m = vx*t

vx = 134m / t

Substitute for vy then vx:

21m = vx * tan 40° * t + 1/2 * -10m/s² * t²

21m = 134m / t * tan 40° * t + 1/2 * -10m/s² * t²

The t cancel out:

21m - 134m*tan 40° = 1/2 * -10m/s² * t²

t = ((21m - 134m*tan 40°) / (1/2 * -10m/s²))

t = 4.28 s

a)

We can use the constant x component of the velocity to calculate the initial velocity.

The relation between the velocity and its x component is:

cos 40° = vx / velocity

vx = dx / t = 134m / 4.28 s = 31.33 m/s

velocity = vx / cos 40° = 40.9 m/s

c)

vx is constant at 31.33 m/s

vy can be calculated using the equation:

v = v0 + a*t

Initial vy = 40.9 m/s * sin 40° = 26.29 m/s

Final vy = 26.29 m/s + (-10m/s²) * 4.28s Final vy = -16.5 m/s

The speed when the ball reaches the wall is (using Pythagorean Theorem):

v = (vx² + vy²) = ((31.33 m/s)² + (-16.5 m/s)²)

v = 35.4 m/s

Answers:

a) Initial ball speed: 40.9 m/s

b) Time for ball to reach wall: 4.28 s

c) x velocity-component: 31.33 m/s

y velocity-component: -16.5 m/s

Speed of the ball: 35.4 m/s

Not sure for height

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