A particle is projected horizontally from top of an incline which makes 37degrees with horizontal. at the same time a flat trolley is released from a point 27m down the incline which goes down without any friction starting from rest. if the particle lands in the trolley find the speed in (m/s) with which the particle is projected.

To solve this problem, we need to analyze the motion of both the particle and the trolley. The key is to determine the time it takes for both the particle and the trolley to reach the same point at the bottom of the incline. Once we have that, we can find the horizontal speed of the particle required for it to land in the trolley.

Understanding the Motion of the Trolley

The trolley is released from rest and moves down the incline under the influence of gravity. Since the incline makes an angle of 37 degrees with the horizontal, we can calculate the acceleration of the trolley using the formula:

• Acceleration (a): a = g * sin(θ)

Here, g is the acceleration due to gravity (approximately 9.81 m/s²), and θ is the angle of the incline (37 degrees). Plugging in the values:

• a = 9.81 * sin(37°) ≈ 9.81 * 0.6018 ≈ 5.91 m/s²

Calculating the Time Taken by the Trolley

Next, we need to find the time it takes for the trolley to travel 27 meters down the incline. We can use the equation of motion:

• Distance (s): s = ut + (1/2)at²

Since the trolley starts from rest, the initial velocity (u) is 0. Thus, the equation simplifies to:

• s = (1/2)at²

Substituting the known values:

• 27 = (1/2)(5.91)t²

Solving for t²:

• 27 = 2.955t²
• t² = 27 / 2.955 ≈ 9.13
• t ≈ √9.13 ≈ 3.02 seconds

Analyzing the Particle's Motion

Now that we have the time it takes for the trolley to reach the bottom, we can analyze the motion of the particle projected horizontally. The horizontal distance traveled by the particle must equal the horizontal distance covered by the trolley in the same time.

During the time t, the particle will fall vertically due to gravity. The vertical distance (h) it falls can be calculated using:

• Vertical Distance (h): h = (1/2)gt²

Substituting the values:

• h = (1/2)(9.81)(3.02)² ≈ (1/2)(9.81)(9.12) ≈ 44.73 meters

However, we need to find the horizontal distance covered by the particle. The horizontal distance (d) can be expressed as:

• d = vt

Where v is the horizontal speed of the particle. The horizontal distance covered by the trolley can be found using the incline angle:

• Horizontal Distance (d_trolley): d_trolley = 27 * cos(37°) ≈ 27 * 0.7986 ≈ 21.56 meters

Finding the Required Speed of the Particle

Now we can set the horizontal distance of the particle equal to the horizontal distance of the trolley:

• vt = d_trolley

Substituting the known values:

• v(3.02) = 21.56
• v = 21.56 / 3.02 ≈ 7.13 m/s

Thus, the speed with which the particle is projected horizontally is approximately 7.13 m/s.