You push a skateboard so that it rolls down the road at a speed of 0.700 m/s. You run after the skateboard at a speed of 3.90 m/s and while still behind the skateboard you jump off the ground at an angle of 28.0° above the horizontal hoping to land on the skateboard. How much distance do you need between you and the skateboard to jump and land on it?

To find the distance you need between you and the skateboard to jump and land on it, we can break down the problem into two components: horizontal and vertical. Let's calculate each component separately.

1. Horizontal Component:
The horizontal component of your velocity remains constant at 3.90 m/s as you run and jump. This means that the skateboard is also moving horizontally at the same speed. The time it takes for you to land on the skateboard horizontally can be calculated using the formula:
Time = Distance / Speed

Since the skateboard is already moving at 0.700 m/s and you are running at 3.90 m/s, the effective relative speed between you and the skateboard is:
Relative Speed = Your Speed - Skateboard Speed
Relative Speed = 3.90 m/s - 0.700 m/s = 3.20 m/s

Now, using the relative speed, we can find the time it takes for you to cover the horizontal distance:
Time = Distance / Relative Speed
Rearranging the equation, we get:
Distance = Relative Speed * Time

The horizontal distance you need to cover is the distance covered by the skateboard during the time you were in the air. This horizontal distance is the product of the relative speed and the time it takes you to land. We can calculate the time using the vertical motion equation:
Vertical Displacement = Initial Vertical Velocity * Time + (1/2) * Acceleration * Time^2

2. Vertical Component:
In the vertical motion, the initial vertical velocity can be calculated using the initial velocity formula:
Vertical Velocity = Initial Velocity * sin(Theta)
Here, Theta is the angle at which you jump, which is 28.0° in this case.

The vertical displacement is the height at which you want to land on the skateboard. Since you jump off the ground, the initial vertical displacement is zero.

The acceleration is due to gravity and is approximately -9.8 m/s^2, considering downward as the positive direction.

Using the vertical motion equation mentioned earlier, we can calculate the time it takes for you to reach the skateboard vertically.

Once we have the time, we can substitute it into the horizontal distance formula to find the distance you need to jump.

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