To solve this problem, we'll need to use the principles of physics, specifically Newton's laws and the equations of motion. Let's break it down step by step.
- Acceleration of the block: The acceleration of the block can be calculated using the component of gravity acting along the inclined plane. The force component along the incline is given by:
F_parallel = m * g * sin(theta)
where m is the mass of the block, g is the acceleration due to gravity (approximately 9.8 m/s^2), and theta is the angle of the incline (25 degrees in this case).
The force parallel to the incline is related to the acceleration through Newton's second law:
F_parallel = m * a
By equating the two expressions for the force parallel to the incline, we can find the acceleration:
m * a = m * g * sin(theta) a = g * sin(theta)
Plugging in the values:
a = 9.8 m/s^2 * sin(25 degrees) a ≈ 4.1 m/s^2
Therefore, the acceleration of the block is approximately 4.1 m/s^2.
- Distance traveled up the incline: To calculate the distance traveled up the incline, we can use the kinematic equation:
v^2 = u^2 + 2 * a * s
where v is the final velocity (which is zero when the block comes to a stop), u is the initial velocity (14 m/s), a is the acceleration (-4.1 m/s^2, since it's acting in the opposite direction to the motion), and s is the distance traveled.
Rearranging the equation to solve for s:
s = (v^2 - u^2) / (2 * a)
Plugging in the values:
s = (0 - 14^2) / (2 * (-4.1)) s ≈ 100.85 m
Therefore, the block will travel approximately 100.85 meters up the incline.
- Time taken to stop: To calculate the time taken for the block to come to a stop, we can use the kinematic equation:
v = u + a * t
where v is the final velocity (0 m/s), u is the initial velocity (14 m/s), a is the acceleration (-4.1 m/s^2), and t is the time taken.
Rearranging the equation to solve for t:
t = (v - u) / a
Plugging in the values:
t = (0 - 14) / (-4.1) t ≈ 3.41 s
Therefore, it will take approximately 3.41 seconds for the block to come to a stop.
To summarize:
- The acceleration of the block is approximately 4.1 m/s^2.
- The block will travel approximately 100.85 meters up the incline.
- It will take approximately 3.41 seconds for the block to come to a stop.