Question

You use a slingshot to launch a potato horizontally from the edge of a cliff with speed v0. The acceleration due to gravity is g. Take the origin at the launch point. Suppose that +y-axis is directed upward and speed v0 is in the +x-direction.

a)How long after you launch the potato has it moved as far horizontally from the launch point as it has moved vertically? Express your answer in terms of some or all of the variables v0 and g.
b) What are the coordinates of the potato at the time it has moved as far horizontally from the launch point as it has moved vertically?
Enter the x and y coordinates separated by a comma. Express your answers in terms of some or all of the variables v0 and g.
c) How long after you launch the potato is it moving in a direction exactly 45∘ below the horizontal?
Express your answer in terms of some or all of the variables v0 and g.
d) What are the coordinates of the potato at the time it is moving in a direction exactly 45∘ below the horizontal?
Enter the x and y coordinates separated by a comma. Express your answers in terms of some or all of the variables v0 and g.

Answer 1

The time it takes for the potato to move as far horizontally as it has vertically can be calculated using the equations of motion. The coordinates of the potato at this time can also be determined using the same equations. The time at which the potato is moving in a direction exactly 45 degrees below the horizontal can be found using trigonometry, and the coordinates at this time can be calculated using the equations of motion.

Step-by-step explanation:

a) The time it takes for the potato to move as far horizontally as it has vertically can be determined using the equations of motion. In the vertical direction, the potato will experience a constant acceleration due to gravity. The time it takes for the potato to reach its maximum height is given by the equation:
t = (v0 sin(θ)) / g, where v0 is the initial velocity and θ is the launch angle.
The time it takes for the potato to reach the same height from its peak can be calculated as twice the time it takes to reach the peak:
tpeak = 2 * t
In the horizontal direction, there is no acceleration, so the time it takes for the potato to move horizontally is the same as the time it takes for it to reach its peak. Therefore, the total time it takes for the potato to move as far horizontally as it has vertically is:
ttotal = t + tpeak

b) The coordinates of the potato at the time it has moved as far horizontally from the launch point as it has moved vertically can be found using the equations of motion. In the horizontal direction, the position is given by:
x = v0 cos(θ) * t
In the vertical direction, the position is given by:
y = v0 sin(θ) * t - 0.5 * g * t2
Substituting the values of t and tpeak calculated in part a, we can find the x and y coordinates.

c) To determine the time at which the potato is moving in a direction exactly 45 degrees below the horizontal, we can use the trigonometric ratio:
tan(θ) = (vy - v0) / vx
Since the initial velocity is in the positive x direction and the angle below the horizontal is 45 degrees, we can solve for t to find the time at which the potato is moving in this direction.

d) To find the coordinates of the potato at the time it is moving in a direction exactly 45 degrees below the horizontal, we can use the equations of motion. In the horizontal direction, the position is given by the equation:
x = v0 cos(θ) * t
In the vertical direction, the position is given by the equation:
y = v0 sin(θ) * t - 0.5 * g * t2
Substituting the values of t calculated in part c, we can find the x and y coordinates.

Answer 2

The time it takes for the potato to move as far horizontally as it has vertically is t = 2v0/g. The coordinates of the potato at this time are x = v0 × (2v0/g) and y = -2v0²/g. The time it takes for the potato to be moving in a direction exactly 45 degrees below the horizontal is t = 2v0/g and the coordinates of the potato at this time are x = v0 × (2v0/g) and y = -2v0.

Step-by-step explanation:

To find the time it takes for the potato to move as far horizontally as it has vertically, we need to determine the horizontal distance traveled and the vertical distance traveled by the potato. The horizontal distance can be found using the formula:

Horizontal distance = initial horizontal velocity × time

Since the potato is launched horizontally, the initial horizontal velocity is equal to the initial speed, v0. The vertical distance can be calculated using the formula:

Vertical distance = 0.5 × g × time²

For the potato to move as far horizontally as it has vertically, the horizontal distance must be equal to the vertical distance. Setting the two equations equal to each other, we get:

v0 × time = 0.5 × g × time²

Dividing both sides of the equation by time, we find:

v0 = 0.5 × g × time

Therefore, the time it takes for the potato to move as far horizontally as it has vertically is t = 2v0/g.

The coordinates of the potato at this time can be calculated using the equations for horizontal and vertical displacement:

Horizontal displacement = initial horizontal velocity × time = v0 × (2v0/g)

Vertical displacement = -0.5 × g × time² = -0.5 × g × (2v0/g)² = -2v0²/g

Thus, the coordinates of the potato at this time are x = v0 × (2v0/g) and y = -2v0²/g.

For the potato to be moving in a direction exactly 45 degrees below the horizontal, the ratio of the vertical velocity to the horizontal velocity must be equal to the tangent of 45 degrees, which is 1. The vertical velocity is given by:

Vertical velocity = initial vertical velocity - g × time = 0 - g × time = -g × t = -g × (2v0/g) = -2v0.

The horizontal velocity remains constant throughout the motion and is equal to the initial horizontal velocity, v0. Therefore, the coordinates of the potato at this time are x = v0 × (2v0/g) and y = -2v0.