Question(II) A block is given an initial speed of $5.2 \mathrm{~m} / \mathrm{s}$ up the $22.0^{\circ}$ plane shown in Fig. 4-45. (a) How far up the plane will it go? (b) How much time elapses before it returns to its starting point? Ignore friction.

Studdy Solution

STEP 1

1. The block is moving on a frictionless inclined plane.
2. The initial speed of the block is $5.2 \, \text{m/s}$ .
3. The angle of the inclined plane is $22.0^\circ$ .
4. We need to find the distance the block travels up the plane and the time taken to return to the starting point.

STEP 2

1. Analyze the motion of the block along the inclined plane.
2. Calculate the distance traveled up the plane.
3. Calculate the time taken to reach the highest point.
4. Calculate the total time for the round trip.

STEP 3

Analyze the motion of the block along the inclined plane.
The block is moving up the plane against gravity. The component of gravitational acceleration acting along the plane is $g \sin(\theta)$ , where $g = 9.8 \, \text{m/s}^2$ and $\theta = 22.0^\circ$ .

STEP 4

Calculate the distance traveled up the plane.
Use the kinematic equation for motion with constant acceleration:
$v^2 = u^2 + 2as$
where $v = 0 \, \text{m/s}$ (final velocity at the highest point), $u = 5.2 \, \text{m/s}$ (initial velocity), $a = -g \sin(\theta)$ (acceleration), and $s$ is the distance traveled.
$0 = (5.2)^2 + 2(-9.8 \sin(22.0^\circ))s$
Solve for $s$ :
$s = \frac{(5.2)^2}{2 \times 9.8 \times \sin(22.0^\circ)}$

STEP 5

Calculate the time taken to reach the highest point.
Use the kinematic equation:
$v = u + at$
where $v = 0 \, \text{m/s}$ , $u = 5.2 \, \text{m/s}$ , and $a = -9.8 \sin(22.0^\circ)$ .
Solve for $t$ :
$0 = 5.2 + (-9.8 \sin(22.0^\circ))t$
$t = \frac{5.2}{9.8 \sin(22.0^\circ)}$

STEP 6

Calculate the total time for the round trip.
The time to go up is equal to the time to come down, so the total time is:
$\text{Total time} = 2t = 2 \times \frac{5.2}{9.8 \sin(22.0^\circ)}$
The distance the block travels up the plane is:
$s = \frac{(5.2)^2}{2 \times 9.8 \times \sin(22.0^\circ)}$
The total time for the round trip is:
$\text{Total time} = 2 \times \frac{5.2}{9.8 \sin(22.0^\circ)}$

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms