To solve this problem, we need to determine the required initial speed for the toy car as it leaves the ramp to ensure it remains in the air for 1.0 seconds. Since the car is launched at an angle, we’ll use the concepts of projectile motion, particularly focusing on the vertical component of the motion.

Steps to Solution

1. Identify known values:

   • Angle of the ramp, $\theta = 32^\circ$
   • Total time of flight, $t = 1.0 \, \text{s}$
   • Acceleration due to gravity, $g = 9.8 \, \text{m/s}^2$

2. Vertical component analysis: For the car to remain in the air for 1 second, it must take $0.5$ seconds to reach its peak and $0.5$ seconds to descend back to the same vertical level. Thus, the vertical velocity component $v_y$ at launch can be calculated from:

   $v_y = g \times t_{up} = 9.8 \times 0.5 = 4.9 \, \text{m/s}$

3. Find the initial speed $v$: The vertical component $v_y$ of the initial velocity $v$ is given by:

   $v_y = v \sin \theta$

   Rearranging to solve for $v$:

   $v = \frac{v_y}{\sin \theta} = \frac{4.9}{\sin 32^\circ}$

4. Calculate $v$: Using $\sin 32^\circ \approx 0.5299$,

   $v = \frac{4.9}{0.5299} \approx 9.2 \, \text{m/s}$

Answer

The correct answer is: $\boxed{9.2 \, \text{m/s}}$

Would you like further details on this solution or have any questions?

Here are five related questions to further your understanding:

1. How would the initial speed change if the ramp angle were increased?
2. What would happen to the required speed if the car needs to be in the air for 2 seconds?
3. How does gravity affect the horizontal component of projectile motion?
4. How would air resistance impact this calculation?
5. Can we calculate the horizontal distance the car travels based on the given data?

Tip: In projectile motion, the vertical and horizontal components of motion are independent of each other. This separation allows us to solve complex motion in simpler parts.