A basketball is thrown horizontally with an initial speed of 4.70m/s . A straight line drawn from the release point to the landing point makes an angle of 30.0 ∘ with the horizontal. What was the release height?

1.5m

To determine the release height of the basketball, we need to understand the motion of the ball and make use of some physics principles.

1. Analyze the motion: We are given that the basketball is thrown horizontally, which means it has no initial vertical velocity. However, the ball will experience a downward acceleration due to gravity.

2. Break down the motion into horizontal and vertical components: The horizontal component of the motion remains constant throughout the entire trajectory, while the vertical component is affected by gravity.

3. Calculate the time of flight: Since there is no initial vertical velocity, we can calculate the time it takes for the ball to reach the ground. We can use the equation:
time = (2 * vertical displacement) / (acceleration due to gravity).

4. Determine the vertical displacement: The vertical displacement is the change in height from the release point to the landing point. To find this, we need to calculate the vertical component of the initial velocity.

5. Calculate the vertical component of initial velocity: The vertical component of velocity is given by the equation:
vertical velocity = initial velocity * sin(angle).

6. Use the time of flight and the vertical component of the initial velocity to solve for the vertical displacement using the equation:
vertical displacement = vertical velocity * time + (1/2) * acceleration due to gravity * time^2.

7. Subtract the vertical displacement from the landing height to find the release height.

Let's plug in the given values and calculate the release height.

Given:
Initial speed (horizontal velocity) = 4.70 m/s
Angle with the horizontal = 30.0°

Acceleration due to gravity is approximately 9.8 m/s^2.

1. Calculate the vertical component of initial velocity:
vertical velocity = 4.70 m/s * sin(30°) = 4.70 m/s * 0.5 = 2.35 m/s.

2. Calculate the time of flight:
time = (2 * vertical displacement) / acceleration due to gravity.

However, we need to find the vertical displacement first.

3. Determine the vertical displacement:
Using the vertical velocity and the time of flight, we can calculate the vertical displacement:
vertical displacement = 2.35 m/s * time + (1/2) * (9.8 m/s^2) * (time^2).

4. Substituting the expression for time from step 2 into the equation for vertical displacement, we have:
vertical displacement = 2.35 m/s * [(2 * vertical displacement) / (9.8 m/s^2)] + (1/2) * (9.8 m/s^2) * [(2 * vertical displacement) / (9.8 m/s^2)]^2.

Simplifying this equation, we get:
1 = 2.35 / 9.8 + (1/2) * [(2 / 9.8) * (vertical displacement)].

Solving for vertical displacement, we find:
vertical displacement = 4.625 meters.

5. Calculate the release height:
The release height is the difference between the landing height and the vertical displacement, so:
release height = landing height - vertical displacement.

By substituting the known values, you can find the release height in your specific scenario.