A box is sent up a frictionless inclined (34.2 degrees) plane at an initial speed 3.9 m/s.
(a)
How much time, in seconds, does it take for the box to stop?
(b)
How far did the box travel up the plane before it came to a stop?
To find the time it takes for the box to stop, we need to use the equations of motion.
First, let's resolve the initial velocity into its components parallel and perpendicular to the inclined plane. The component parallel to the plane is given by v_parallel = v_initial * cos(theta), where theta is the angle of incline (34.2 degrees) and v_initial is the initial speed (3.9 m/s). Therefore, v_parallel = 3.9 m/s * cos(34.2 degrees).
Since the inclined plane is frictionless, the only force acting on the box is the component of the gravitational force parallel to the plane, which can be calculated as F_parallel = m * g * sin(theta), where m is the mass of the box and g is the acceleration due to gravity (approximately 9.8 m/s^2). We can express this force as F_parallel = m * a, where a is the acceleration of the box parallel to the plane.
Applying Newton's second law (F_parallel = m * a), we get m * a = m * g * sin(theta). Here, mass cancels out, and we are left with a = g * sin(theta).
Next, we can use the equation of motion v_final = v_initial + a * t, where v_final is the final velocity of the box (which will be zero since it stops), t is the time taken to stop, and a is the acceleration. Rearranging the equation, we have t = (v_final - v_initial) / a.
Let's calculate the time it takes for the box to stop:
v_final = 0 m/s (since the box comes to stop)
v_initial = 3.9 m/s * cos(34.2 degrees)
a = g * sin(theta)
Now we can substitute these values into the equation and calculate the time:
t = (0 m/s - (3.9 m/s * cos(34.2 degrees))) / (g * sin(34.2 degrees))
To find the distance traveled by the box up the inclined plane before coming to a stop, we can use the equation of motion s = v_initial * t + 0.5 * a * t^2, where s is the distance traveled.
Substituting the values for v_initial and a, and using the calculated value of t, we can find the distance traveled.
s = (3.9 m/s * cos(34.2 degrees)) * t + 0.5 * (g * sin(34.2 degrees)) * t^2
Let's calculate the distance traveled:
s = (3.9 m/s * cos(34.2 degrees)) * t + 0.5 * (g * sin(34.2 degrees)) * t^2