Question
This is SS 2 physics question answer according to the class
A tennis ball projected at an angle theta attains a range of r equals to 78 m if the velocity impacted on the ball by the rocket is 30 m per second calculate the angle of theta take g as 10 m/s
A tennis ball projected at an angle theta attains a range of r equals to 78 m if the velocity impacted on the ball by the rocket is 30 m per second calculate the angle of theta take g as 10 m/s
Answers
GPT-4o mini
To solve for the angle \(\theta\) of the tennis ball projected at an angle that achieves a range of \(R = 78 \, m\) with an initial velocity \(v = 30 \, m/s\), we can use the formula for the range of a projectile:
\[
R = \frac{v^2 \sin(2\theta)}{g}
\]
where:
- \(R\) is the range,
- \(v\) is the initial velocity,
- \(g\) is the acceleration due to gravity.
Given:
- \(R = 78 \, m\)
- \(v = 30 \, m/s\)
- \(g = 10 \, m/s^2\)
We can substitute these values into the formula and solve for \(\sin(2\theta)\):
\[
78 = \frac{(30)^2 \sin(2\theta)}{10}
\]
First, simplify the equation:
\[
78 = \frac{900 \sin(2\theta)}{10}
\]
\[
78 = 90 \sin(2\theta)
\]
Next, divide both sides by 90:
\[
\sin(2\theta) = \frac{78}{90} = \frac{13}{15}
\]
Now, we need to find \(2\theta\):
To find the angle \(2\theta\), we can use the inverse sine function:
\[
2\theta = \sin^{-1}\left(\frac{13}{15}\right)
\]
Calculating \(2\theta\):
Using a calculator,
\[
2\theta \approx 0.837 \, \text{radians} \quad \text{(Convert to degrees if needed: } \approx 48.0^\circ\text{)}
\]
Finally, to find \(\theta\):
\[
\theta = \frac{2\theta}{2} = \frac{0.837}{2} \approx 0.419 \, \text{radians}
\]
Convert to degrees:
\[
\theta \approx 0.419 \times \frac{180}{\pi} \approx 24.0^\circ
\]
Thus, the angle \(\theta\) is approximately:
\[
\theta \approx 24^\circ
\]
\[
R = \frac{v^2 \sin(2\theta)}{g}
\]
where:
- \(R\) is the range,
- \(v\) is the initial velocity,
- \(g\) is the acceleration due to gravity.
Given:
- \(R = 78 \, m\)
- \(v = 30 \, m/s\)
- \(g = 10 \, m/s^2\)
We can substitute these values into the formula and solve for \(\sin(2\theta)\):
\[
78 = \frac{(30)^2 \sin(2\theta)}{10}
\]
First, simplify the equation:
\[
78 = \frac{900 \sin(2\theta)}{10}
\]
\[
78 = 90 \sin(2\theta)
\]
Next, divide both sides by 90:
\[
\sin(2\theta) = \frac{78}{90} = \frac{13}{15}
\]
Now, we need to find \(2\theta\):
To find the angle \(2\theta\), we can use the inverse sine function:
\[
2\theta = \sin^{-1}\left(\frac{13}{15}\right)
\]
Calculating \(2\theta\):
Using a calculator,
\[
2\theta \approx 0.837 \, \text{radians} \quad \text{(Convert to degrees if needed: } \approx 48.0^\circ\text{)}
\]
Finally, to find \(\theta\):
\[
\theta = \frac{2\theta}{2} = \frac{0.837}{2} \approx 0.419 \, \text{radians}
\]
Convert to degrees:
\[
\theta \approx 0.419 \times \frac{180}{\pi} \approx 24.0^\circ
\]
Thus, the angle \(\theta\) is approximately:
\[
\theta \approx 24^\circ
\]