A carpenter asks his helper to toss an apple to him. The carpenter is 12 ft up and 12 ft to the right of the helper, who throws the apple at Vo=32 ft/s.
There are two values of theta that cause that apple to land in the carpenters hand. What are they?
To find the two possible values of theta that cause the apple to land in the carpenter's hand, we can use projectile motion equations.
First, we need to break down the initial velocity (Vo) into its horizontal (Vox) and vertical (Voy) components. Voy represents the vertical velocity, and Vox represents the horizontal velocity.
Given:
Vo = 32 ft/s
The apple is thrown from the helper to the carpenter, who is 12 ft up and 12 ft to the right.
Vertical motion:
1. Displacement (Δy) = 12 ft (the vertical distance between the helper and the carpenter)
2. Initial vertical velocity (Voy) = ? (Unknown)
3. Initial vertical position (y0) = 0 (since the helper is at the height of the carpenter)
Using the equation:
Δy = Voy * t - 0.5 * g * t^2 (where g is the acceleration due to gravity)
We can substitute the given values:
12 = Voy * t - 0.5 * g * t^2
Horizontal motion:
1. Displacement (Δx) = 12 ft (the horizontal distance between the helper and the carpenter)
2. Velocity (Vox) = 32 ft/s (given)
3. Initial horizontal position (x0) = 0 (since the helper is at the starting point)
Using the equation:
Δx = Vox * t
We can substitute the given values:
12 = 32 * t
Simplifying this equation:
12 = 32t
t = 12/32
t = 0.375 seconds
Now, we can substitute the value of t back into the vertical motion equation:
12 = Voy * 0.375 - 0.5 * g * 0.375^2
Simplifying further:
12 = 0.375Voy - 0.0703g (1)
To find the two possible values of theta, we can use the trigonometric relationship between the horizontal and vertical velocities:
tan(theta) = Voy / Vox
We know the values of Voy and Vox, so we can substitute them into the equation:
tan(theta) = Voy / 32
Now, we need to solve these equations simultaneously to find the two values of theta.
To solve Equation (1):
12 = 0.375Voy - 0.0703g
And, tan(theta) = Voy / 32
We can rearrange Equation (1) to get Voy in terms of g:
Voy = (12 + 0.0703g) / 0.375
Substituting this value of Voy into the second equation:
tan(theta) = (12 + 0.0703g) / (0.375 * 32)
Now, we can solve for g:
g = 32^2 * tan(theta) - 12 / 0.0703
Using a numerical method or calculator, you can find the value of g. Substituting that value back into the equation will give you the two possible values of theta.
Please note that the exact values of theta will depend on the specific acceleration due to gravity at that location.