1. a car is driven east for adistance of52 km,then north for 33 km, and then in a direction32 degree east of north for 28 km.draw the vector diagram and determine the total displacement of the car from its starting point. and direction(counterclockwise from east. Q2. a plane diving with constant speed at an angle of 47 degree with the vertical, releases a projectile at an altitude of 540 m. the projectile hits the ground 5 s after release.(assume a coordinate system in which the plane is moving in the positive horizontal direction, and in negative vertical direction. no air resistance. (a) what is the speed of air craft? (b) how far did the projectile travel horizontally during its flight? (c) what were the horizontal and vertical components of its velocity just before striking the ground?

Let's break down your questions step by step, starting with the car's journey and then moving on to the plane and projectile. We’ll use vector diagrams and some basic physics principles to find the answers.

Analyzing the Car's Journey

The car travels in three segments: first east, then north, and finally at an angle. To find the total displacement, we can represent each segment as a vector and then add them together.

Step 1: Representing the Journey

• First Segment: 52 km east can be represented as the vector (52, 0).
• Second Segment: 33 km north can be represented as the vector (0, 33).
• Third Segment: 28 km at 32 degrees east of north requires some trigonometry:
  • Horizontal component: 28 * sin(32°)
  • Vertical component: 28 * cos(32°)

Step 2: Calculating the Third Segment

Using a calculator, we find:

• Horizontal component: 28 * sin(32°) ≈ 14.6 km
• Vertical component: 28 * cos(32°) ≈ 24.0 km

Step 3: Summing the Vectors

Now we can sum the vectors:

• Total horizontal displacement = 52 km + 14.6 km = 66.6 km
• Total vertical displacement = 33 km + 24.0 km = 57.0 km

Step 4: Finding the Total Displacement

The total displacement can be found using the Pythagorean theorem:

Total displacement = √(66.6² + 57.0²) ≈ √(4436.36 + 3249) ≈ √7685.36 ≈ 87.7 km

Step 5: Determining the Direction

To find the angle counterclockwise from east, we use the tangent function:

θ = tan⁻¹(vertical displacement / horizontal displacement) = tan⁻¹(57.0 / 66.6) ≈ 40.5 degrees

Exploring the Plane and Projectile Motion

Now, let's analyze the plane releasing a projectile. We’ll break this down into parts as well.

Part (a): Speed of the Aircraft

The projectile is released from an altitude of 540 m and takes 5 seconds to hit the ground. We can find the vertical velocity using the formula:

Vertical distance = initial vertical velocity * time + 0.5 * g * time²

Here, g (acceleration due to gravity) is approximately 9.81 m/s². Since the projectile starts from rest vertically, the initial vertical velocity is 0:

540 m = 0 + 0.5 * 9.81 m/s² * (5 s)²

540 m = 0.5 * 9.81 * 25

540 m = 122.625 m

Now, we can find the vertical velocity just before hitting the ground:

Vertical velocity = g * time = 9.81 m/s² * 5 s = 49.05 m/s

Since the plane is diving at an angle of 47 degrees with the vertical, we can find the speed of the aircraft:

Vertical velocity = speed * cos(47°)

Thus, speed = vertical velocity / cos(47°) ≈ 49.05 / 0.681 = 72.0 m/s

Part (b): Horizontal Distance Traveled

The horizontal distance can be calculated using the horizontal component of the velocity:

Horizontal velocity = speed * sin(47°) ≈ 72.0 * 0.731 = 52.7 m/s

Distance = horizontal velocity * time = 52.7 m/s * 5 s = 263.5 m

Part (c): Components of Velocity Before Impact

Just before striking the ground, the horizontal and vertical components of the projectile's velocity are:

• Horizontal Component: 52.7 m/s
• Vertical Component: 49.05 m/s

In summary, we’ve calculated the total displacement of the car and analyzed the projectile motion of the plane. Each step involved breaking down the problem into manageable parts and applying relevant physics principles. If you have any further questions or need clarification on any part, feel free to ask!