The problem provided involves projectile motion, specifically asking to determine two possible angles of projection $\theta_1$ and $\theta_2$ such that the projectile hits a target 91.4 meters away, given the initial velocity $v_0 = 427 \, \text{m/s}$. It also asks to compute the time it takes for the projectile to reach the target for each angle.

Let's break down the two parts:

Part a: Determining the angles of projection ($\theta_1$ and $\theta_2$)

For projectile motion, the horizontal range $R$ is given by the formula: $R = \frac{v_0^2 \sin(2\theta)}{g}$ where:

• $R = 91.4 \, \text{m}$ is the range,
• $v_0 = 427 \, \text{m/s}$ is the initial velocity,
• $g = 9.8 \, \text{m/s}^2$ is the acceleration due to gravity,
• $\theta$ is the angle of projection.

Rearranging this equation to solve for $\theta$: $\sin(2\theta) = \frac{R \cdot g}{v_0^2}$

Substitute the known values: $\sin(2\theta) = \frac{91.4 \times 9.8}{427^2}$

We can solve for $2\theta$ and then find $\theta_1$ and $\theta_2$, as $\sin(2\theta)$ will give two possible angles.

Part b: Time of flight for each angle

The time of flight $T$ for a projectile is given by: $T = \frac{2v_0 \sin(\theta)}{g}$ We'll calculate the time for both $\theta_1$ and $\theta_2$ obtained in part (a).

I'll calculate these now.The two possible angles of projection are:

• $\theta_1 \approx 0.14^\circ$
• $\theta_2 \approx 89.86^\circ$

For the time of flight:

• At $\theta_1 \approx 0.14^\circ$, the time of flight is approximately 0.214 seconds.
• At $\theta_2 \approx 89.86^\circ$, the time of flight is approximately 87.14 seconds.

Would you like more details or have any questions? Here are some follow-up questions to deepen understanding:

1. Why are there two possible angles for the same target range?
2. How does the angle affect the time of flight in projectile motion?
3. Can you explain how trigonometric identities were used in this solution?
4. What would happen to the angles if the target were further away?
5. How does changing the initial velocity affect the angles and time?

Tip: Always remember that in projectile motion, a higher angle gives a higher trajectory, but it also increases the time of flight significantly!