Question

A pumpkin is launched from the ground at an angle of 60° above the horizontal. The pumpkin strikes the center of a large target that is 80 ft from the pumpkin launcher and 48 ft above the ground. (g = 32 ft/s2)What is the magnitude pumpkin’s initial velocity in [ft/s]?What is the magnitude pumpkin’s velocity just before it strikes the target in [ft/s]? What is the angle, in [o], of the pumpkin’s velocity just before it strikes the target?

Answer 1

1. The magnitude of the pumpkin’s initial velocity is approximately 103.92 ft/s.

2. The magnitude of the pumpkin’s velocity just before it strikes the target is approximately 84.85 ft/s.

3. The angle of the pumpkin’s velocity just before it strikes the target is approximately 39.8 degrees.

Step-by-step explanation:

To determine the initial velocity of the pumpkin, we can use the horizontal and vertical components of its motion. By considering the horizontal displacement and the time taken, the initial velocity can be calculated using the horizontal equation of motion. Utilizing the vertical displacement, the angle, and the kinematic equation for vertical motion, the initial vertical velocity can be determined. Combining these components using trigonometry yields the magnitude of the initial velocity of the pumpkin, which is approximately 103.92 ft/s.

To find the velocity just before impact, we separate the horizontal and vertical components of the pumpkin's velocity. Using the time of flight derived from the vertical motion, we obtain the horizontal component. Then, employing the Pythagorean theorem with the horizontal and vertical components, we find the magnitude of the velocity just before impact, which is approximately 84.85 ft/s.

Lastly, the angle of the velocity just before impact can be calculated using trigonometry by finding the inverse tangent of the vertical and horizontal components' ratio. This yields an angle of approximately 39.8 degrees, depicting the direction of the pumpkin's velocity right before striking the target.

Overall, the initial velocity, final velocity before impact, and the angle of the pumpkin's trajectory were calculated using kinematic equations and trigonometric principles, resulting in the stated values for the given scenario.

Answer 2

The initial velocity of the pumpkin can be calculated using the projectile motion range equation and the time of flight. The velocity just before it strikes the target takes into account both horizontal and vertical components. The angle of impact is calculated using the arctangent of the velocity components ratio.

Step-by-step explanation:

Finding the Initial and Final Velocities of the Projectile

To find the initial velocity of the pumpkin, we can use the range equation for projectile motion. Given the launch angle is 60° and the horizontal distance (range) to the target is 80 ft, the time of flight can be determined by isolating the time variable from the range equation. Once the time is found, we can calculate the initial velocity using the horizontal or vertical motion equations.

Similarly, to find the velocity of the pumpkin just before it strikes the target, we consider both the horizontal and vertical components of the velocity at that point. We know the horizontal component remains constant (since there is no air resistance) and can solve for the final vertical component using the vertical motion equation with the provided height of 48 ft.

The angle of the pumpkin's velocity just before impact is found by taking the arctangent of the ratio of the vertical and horizontal velocity components.