A truck loaded with cannonball watermelons stops suddenly to avoid running over the edge of a washed-out bridge (see figure). The quick stop causes a number of melons to fly off the truck. One melon rolls over the edge with an initial speed vi = 12.0 m/s in the horizontal direction. A cross-section of the bank has the shape of the bottom half of a parabola with its vertex at the edge of the road, and with the equation y2 = 16x, where x and y are measured in meters. What are the x and y coordinates of the melon when it splatters on the bank?

To find the x and y coordinates of the melon when it splatters on the bank, we need to determine the point where its trajectory intersects with the bank.

Let's analyze the motion of the melon. We know that the initial horizontal speed is vi = 12.0 m/s, and the acceleration in the horizontal direction is zero since there is no force acting horizontally to change its velocity.

In the vertical direction, we have a constant downward acceleration due to gravity with a magnitude of g = 9.8 m/s^2.

Based on the information given, the melon's motion can be broken down into horizontal and vertical components.

First, let's find the time it takes for the melon to hit the bank. We can use the equation for vertical displacement:

y = vi*t + (1/2)*g*t^2

Since the melon splatters on the bank when it hits, its vertical displacement, y, will be zero. So we can rewrite the equation as:

0 = vi*t + (1/2)*g*t^2

Simplifying this equation, we get:

(1/2)*g*t^2 = -vi*t

Plugging in the values, we have:

(1/2)*9.8*t^2 = -12*t

Simplifying further:

4.9*t^2 = -12*t

Dividing both sides by t (since t cannot be zero), we get:

4.9*t = -12

t = -12 / 4.9

t ≈ -2.45 seconds

Since time cannot be negative, we discard the negative value. So, the time it takes for the melon to hit the bank is approximately 2.45 seconds.

Next, let's find the horizontal displacement, x, using the equation:

x = vi*t

Plugging in the values, we have:

x = 12.0 * 2.45

x ≈ 29.4 meters

Therefore, the x coordinate of the melon when it splatters on the bank is approximately 29.4 meters.

Now, let's find the y coordinate of the melon when it splatters on the bank. We can use the equation for vertical displacement again:

y = vi*t + (1/2)*g*t^2

Plugging in the values, we have:

y = 12.0 * 2.45 + (1/2)*9.8*(2.45)^2

y ≈ 29.4 + 30.1

y ≈ 59.5 meters

Therefore, the y coordinate of the melon when it splatters on the bank is approximately 59.5 meters.

To summarize:
- The x coordinate of the melon is approximately 29.4 meters.
- The y coordinate of the melon is approximately 59.5 meters.

To solve this problem, we need to find the x and y coordinates of the melon when it splatters on the bank. We are given the initial horizontal speed of the melon, and we know the shape of the bank.

First, let's analyze the motion of the melon. Since there are no external forces acting on the melon in the horizontal direction, its horizontal velocity will remain constant throughout its motion. Therefore, the x-coordinate of the melon will be determined solely by its initial velocity and the time it takes to reach the bank.

To find the time it takes for the melon to reach the bank, we need to consider that the motion of the melon in the vertical direction is influenced by gravity. The equation for the vertical motion of an object near the surface of the Earth is given by:

y = vi*t + (1/2)*g*t^2

where y is the vertical position, vi is the initial vertical velocity, t is the time, and g is the acceleration due to gravity (approximately 9.8 m/s^2).

In this case, the initial vertical velocity vi is zero since the melon is rolling horizontally. The vertical position of the melon when it splatters on the bank is given by the equation of the bank: y^2 = 16x.

By substituting y = √(16x) into the equation of motion in the vertical direction and solving for t, we can find the time it takes for the melon to reach the bank.

√(16x) = (1/2)*g*t^2

Simplifying the above equation gives:

t = √(32x/g)

Now that we have the time it takes to reach the bank, we can calculate the x-coordinate of the melon. Since the initial horizontal velocity vi is constant, the x-coordinate is given by:

x = vi * t

Substituting the expression for t from above into this equation, we get:

x = vi * √(32x/g)

Squaring both sides of the equation to eliminate the square root gives:

x^2 = (vi^2 * 32x) / g

Rearranging the equation gives:

x = (vi^2 * 32) / g

Finally, we can substitute the given initial horizontal velocity vi = 12.0 m/s and the acceleration due to gravity g = 9.8 m/s^2 into this expression to find the value of x:

x = (12.0^2 * 32) / 9.8

Solving this equation will give us the value of x. Once we have x, we can substitute it into the equation of the bank y^2 = 16x to find the corresponding y-coordinate.

Please note that I have assumed that the melon does not experience any air resistance during its motion, and that the surface of the bank is smooth.