A rock is thrown upward from the level ground in such a way that the
maximum height of its flight is equal to its horizontal range R. (a) At what angle is the rock thrown? (b)
Would your answer to part (a) be different if the rock is thrown with the same speed on a different planet?
To find the angle at which the rock is thrown, we can use the principles of projectile motion. Let's solve it step-by-step:
Step 1: Define the variables:
Let's assume:
- Angle at which the rock is thrown = θ
- Initial velocity of the rock = v0
- Gravitational acceleration = g (which is approximately 9.8 m/s² on Earth)
Step 2: Analyze the motion horizontally:
Since the maximum height of the rock's flight is equal to its horizontal range (R), we can write the equation:
R = (v0² * sin(2θ)) / g --------(Equation 1)
Step 3: Analyze the motion vertically:
The equations of motion for the vertical component of the rock's motion are:
v = v0 * sin(θ) - g * t --------(Equation 2)
h = v0 * sin(θ) * t - (1/2) * g * t² --------(Equation 3)
Where:
- v is the vertical velocity of the rock at time t
- h is the height of the rock at time t
Step 4: Find the time of flight (t):
At the maximum height of the rock's flight, the vertical velocity will be zero. So, we can set Equation 2 to zero and solve for t:
0 = v0 * sin(θ) - g * t
t = v0 * sin(θ) / g --------(Equation 4)
Step 5: Substitute the value of t into Equation 3:
Substituting the value of t from Equation 4 into Equation 3, we can find the maximum height (h):
h = (v0² * sin²(θ)) / (2 * g) --------(Equation 5)
Step 6: Equate the maximum height (h) and the horizontal range (R):
We have the Equation 1 for the horizontal range (R) and Equation 5 for the maximum height (h). Equating them, we get:
(v0² * sin(2θ)) / g = (v0² * sin²(θ)) / (2 * g)
Step 7: Simplify and solve for θ:
By canceling out common terms and rearranging, we can find the value of θ:
sin(2θ) = 2 * sin²(θ)
2 * sin(θ) * cos(θ) = 2 * sin²(θ)
cos(θ) = sin(θ)
tan(θ) = 1
θ = 45°
Therefore, the rock is thrown at an angle of 45 degrees.
Now, let's move on to part (b) of the question:
Would your answer to part (a) be different if the rock is thrown with the same speed on a different planet?
The answer is yes. The angle at which the rock is thrown depends on the gravitational acceleration of the planet. If the gravitational acceleration on the different planet is not equal to the value of g on Earth (9.8 m/s²), the angle at which the rock is thrown would be different. To determine the angle at which the rock is thrown on a different planet, you would need to know the gravitational acceleration of that particular planet and repeat the steps outlined above using the new value of g.