Porjectile motion, projectile motion

Question

a baseball leaves the bat of a Mark with a speed of 34m/s at an angle of 37degrees. above the horizontal. the ball is 1.2m off whenit leaves the bat. to ba a home run, the ball must clear a fence 3.0m high and 105m from the batter. will Mark have a home run?

Answer 1

the vertical speed at the start is:

v = 34*sin(37)

v = 20.4617 m/s

time to reach max height will be when the velocity goes to 0:

v = 0 = v0 - gt ... g = acceleration of gravity = 9.8 m/s^2

20.4617 = (9.8)t

t =  2.0879 seconds

maximum height above the ground is:

s = s0 + vt - (1/2)gt^2

s = 1.2 + 20.4617(2.0879) - 4.9(2.0879)^2

s =  22.5613 meters

time to fall back down to 3 meters above the ground:

s = (22.5613 - 3) = (1/2)gt^2 = 4.9t^2

t = SQRT(19.5613/4,9)

t = 1.9980 seconds

So the total travel time from leaving the bat to when the ball will be at 3 meters coming down is:

T = 2.0879 + 1.9980 seconds

T = 4.0859 seconds

How far will the ball travel horizontally in this time?

need the horizontal velocity when the ball left the bat:

u = 34cos(37) = 27.1536

this times the total time will be the distance:

distance = uT = (27.1536)(4.0859) = 110.9469 meters

yes Mark will have a home run since the ball will easily clear the fence.

Answer 2

a baseball leaves the bat of a Mark with a speed of 34m/s at an angle of 37degrees. above the horizontal. the ball is 1.2m off whenit leaves the bat. to ba a home run, the ball must clear a fence 3.0m high and 105m from the batter. will Mark have a home run?

Vo = 34 m/s

a = 37 deg

yo = 1.2 m

fence

y = 3 m

x = 105 m

Compute the x and y components of the velocity

Vox = Vo cos a = 34 cos(37 deg) = 34 * 0.799

Vox = 27.2 m/s

Voy = Vo sin a = 34 sin (37 deg) = 34 * 0.602

Voy = 20.5 m/s

compute the time to cover the distance to the fence

x = Vox t

105 = 27.2 * t

t = 105/27.2

t = 3.86 sec

compute the height of the ball when it is above the fence at

time t = 3.86 sec, the position should be greater than 3 m so that the ball will clear the fence

Δy = Voy t -0.5g t^2

y - 1.2 = 20.5(3.86) - 4.9 (3,86)^2

y = 1.2 + 79.13 - 4.9 (14.9)

y = 7.3 m

7.3 m > 3 m, thus the ball will clear the fence and a....

HOME RUN !!!