A person who is 1.6 m tall throws a ball with a speed of 6.4 m/s at an angle of 65 degrees above the horizontal. What are the x and y components of the velocity (in m/s) of the ball the instant before it strikes the ground?
Hi = 1.6
u = 6.4 cos 65 forever = x coordinate
Vi = 6.4 sin 65 initial
Vertical problem:
v = Vi - 9.81 t
h = Hi + Vi t - 4.9 t^2
so
0 = 1.6 + 5.8 t - 4.9 t^2
or
4.9 t^2 - 5.8 t - 1.6 = 0
t = [ 5.8 +/- sqrt (33.64 + 31.36)]/9.8
t = (5.8 +/- 8.06)/9.8
we want + answer
t = 1.41 seconds in the aair
v = Vi - 9.81 t
v = 5.8 - 9.81 (1.41)
v = -8.08 m/s = y coordinate
To find the x and y components of the velocity of the ball before it strikes the ground, we first need to break down the initial velocity into its horizontal (x) and vertical (y) components.
The horizontal component (Vx) of the velocity is given by Vx = V * cos(θ), where V is the magnitude of the velocity (6.4 m/s) and θ is the angle (65 degrees).
Vx = 6.4 m/s * cos(65 degrees)
Vx ≈ 6.4 m/s * 0.4226
Vx ≈ 2.70 m/s
The vertical component (Vy) of the velocity is given by Vy = V * sin(θ).
Vy = 6.4 m/s * sin(65 degrees)
Vy ≈ 6.4 m/s * 0.9063
Vy ≈ 5.80 m/s
Therefore, the x-component of the velocity is approximately 2.70 m/s, and the y-component of the velocity is approximately 5.80 m/s.
To find the x and y components of the velocity of the ball, we can use some basic trigonometry.
The x component of the velocity is the horizontal component, which remains constant throughout the motion. The y component of the velocity is the vertical component, which changes due to the effect of gravity.
Let's break down the problem step by step:
Step 1: Find the x component of the velocity (Vx):
Given that the ball is thrown at an angle of 65 degrees above the horizontal and assuming no air resistance, we need to find the horizontal component of the initial velocity (Vx).
Vx = V * cos(θ)
Where:
V = initial speed = 6.4 m/s
θ = angle = 65 degrees
Using this formula, we can calculate the x component of the velocity:
Vx = 6.4 m/s * cos(65 degrees)
Vx ≈ -2.759 m/s (since the ball is thrown to the left).
Step 2: Find the y component of the velocity (Vy):
The vertical component of the initial velocity (Vy) changes due to the effect of gravity over time. We need to find the value of Vy just before the ball strikes the ground.
Vy = V * sin(θ) - g * t
Where:
V = initial speed = 6.4 m/s
θ = angle = 65 degrees
g = acceleration due to gravity = 9.8 m/s^2
t = time to reach the ground
To find t, we need to use the kinematic equation for vertical motion:
y = V * sin(θ) * t - 0.5 * g * t^2
Since the ball is thrown vertically downward when it reaches the ground, y = -1.6 m.
-1.6 m = (6.4 m/s) * sin(65 degrees) * t - 0.5 * (9.8 m/s^2) * t^2
Solving this quadratic equation will give us two possible values for t. We discard the negative value since time cannot be negative in this context.
Using the quadratic formula, we get:
t ≈ 1.58 s
Now we can find the y component of the velocity:
Vy = V * sin(θ) - g * t
Vy = (6.4 m/s) * sin(65 degrees) - (9.8 m/s^2) * (1.58 s)
Vy ≈ 0.12 m/s
So, the x component of the velocity is approximately -2.759 m/s, and the y component is approximately 0.12 m/s.