Question

A baseball goes after being hit by a bat with a 40 m/s velocity at a 30° angle. What are the horizontal and vertical components of the velocity after 3 seconds? What are the maximum height and the horizontal reach? ​

Answer 1

Answers:

After 3 seconds:

• Horizontal Component of velocity = 34.6410 m/s
• Vertical Component of velocity = -9.4 m/s

Maximun Height (y) = 20.4082 m

Horizontal Reach (x) = 141.3908 m

Process:

Data:

• v0 = 40 m/s
• θ = 30°
• t = 3 s
• g = -9.8 m/s per second

We start by using a trigonometry funtion to figure out the components of the initial speed:

• Component x = v0 * cos(θ)
• Component y = v0 * sin(θ)

Component x = 40 * cos(30°) = 34.6410 m/s

Component y = 40 * sin (30°) = 20 m/s

We will use the next formula to find the velocity at t = 3s

vf = v0 + a*t

vf(component x) = v0(component x) + a*t

vfx = 34.6410 m/s + (0m/s per second) * (3 s) (You can see that there is no acceleration involving, because the is no air resistance)

vfx = 34.6410 m/s

vf(component y) = v0(component y) + a*t

vfy = 20 m/s + (-9.8m/s per second) * (3s) (We use the gravitational acceleration only for the vertical component)

vfy = 20 m/s - 29.4m/s

vfy = -9.4 m/s

To calculate the maximum height we will say that the vertical velocity is 0 m/s and solve the time variable:

vfy = v0y + a * t

0 m/s = 20 m/s + (-9.8m/s per second) * t

t = (-20 m/s) / (-9.8 m/s per second)

t = 2.0408 s

Now we will evaluate the maximun height with the formula:

y = yo + voy * t + (1/2)*a*t*t

y = 0m + (20m/s)*(2.0408s) + (1/2)(-9.8m/s per second)(2.0408s)(2.0408s)

y = 40.816 m - 20.4078 m

y = 20.4082 m

We will figure out how much time it takes for the ball to hit the floor using the same formula as above, but y = 0

0 = 0m + (20m/s)*t + (1/2)(-9.8m/s per second) * t * t

t ( 20m/s -4.9m/s per second * t)  = 0

t1 = 0 seconds

t2 = (-20m/s) / (-4.9 m/s per second)

t2 = 4.0816 seconds

For the horizontal reach we will use t2 to solve the next formula:

x = x0 + v0x * t + (1/2)*a*t*t

x = 0m + (34.6410m/s)(4.0816s) + (1/2)(0 m/s per second)(4.0816s)(4.0816s)

x = (34.6410 m/s)(4.0816s)

x = 141.3907 m

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