To find the initial speed of the projectile, we need to apply the equations of projectile motion. The key idea is to break the motion into horizontal and vertical components.

Step 1: Break into components

Given dataTo find the initial speed of the projectile, we need to apply the equations of projectile motion. The key idea is to break the motion into horizontal and vertical components.

Step 1: Break into components

Given

• Height of the table: $h = 1.43 \, \text{m}$
• Angle of projection: $\theta = 28^\circ$
• Horizontal distance (range): $R = 5.48 \, \text{m}$
• Acceleration due to gravity: $g = 9.8 \, \text{m/s}^2$

We need to determine the initial speed $v_0$.

Step 2: Horizontal motion equation

The horizontal distance covered is given by: $R = v_0 \cos(\theta) \cdot t$ where $t$ is the time of flight.

Step 3: Vertical motion equation

For vertical motion, the vertical displacement $\Delta y = -1.43 \, \text{m}$ (since the projectile falls below its starting height) is given by: $\Delta y = v_0 \sin(\theta) \cdot t - \frac{1}{2} g t^2$

Substitute $\Delta y = -1.43 \, \text{m}$ and solve the system of equations to find the initial speed $v_0$.

Step 4: Solving for time $t$

From the horizontal equation: $t = \frac{R}{v_0 \cos(\theta)}$

Substitute this into the vertical equation: $-1.43 = v_0 \sin(\theta) \cdot \frac{R}{v_0 \cos(\theta)} - \frac{1}{2} g \left( \frac{R}{v_0 \cos(\theta)} \right)^2$

Now, solve this equation for $v_0$.

Let's compute this step-by-step.The positive solution for the initial speed $v_0$ is approximately $6.59 \, \text{m/s}$.

Thus, the initial speed of the projectile is about 6.59 m/s.

Do you want more details or have any questions?

Here are 5 related questions for further practice:

1. What would be the time of flight for the projectile?
2. How would the initial speed change if the angle of projection were increased?
3. What is the maximum height the projectile reaches during its flight?
4. How does the range of the projectile change if the initial speed is doubled?
5. What would happen to the range if the table height were halved?

Tip: In projectile motion, breaking the motion into horizontal and vertical components simplifies complex problems.