If the back of the truck is 1.5 {\rm m} above the ground and the ramp is inclined at 24{\rm ^{\circ}}, how much time do the workers have to get to the piano before it reaches the bottom of the ramp?
It depends on the friction on the ramp.
To determine how much time the workers have to get to the piano before it reaches the bottom of the ramp, we need to consider the motion of the piano down the ramp.
Let's break down the problem into two parts: the horizontal motion along the ramp and the vertical motion perpendicular to the ramp.
First, let's consider the horizontal motion along the ramp. We need to determine the horizontal distance traveled by the piano while it is rolling down the ramp. We'll use the formula for the horizontal distance:
\[ \text{Horizontal distance} = \text{velocity} \times \text{time} \]
In this case, the velocity along the ramp is a component of the piano's initial velocity, which can be calculated using the formula:
\[ \text{Velocity along ramp} = \text{initial velocity} \times \cos(\text{angle of incline}) \]
The initial velocity of the piano can also be calculated using the formula for the initial horizontal velocity of an object in projectile motion:
\[ \text{Initial velocity} = \text{launch velocity} \times \sin(\text{angle of incline}) \]
The launch velocity is the velocity of the piano just before it starts rolling down the ramp. We don't have this information, so we'll assume it's given or can be measured.
Next, let's consider the vertical motion perpendicular to the ramp. We need to determine the time it takes for the piano to travel in the vertical direction from a height of 1.5 m above the ground to the ground. We'll use the formula for the time of flight of an object in projectile motion:
\[ \text{Time of flight} = \frac{2 \times \text{height}}{\text{vertical component of initial velocity}} \]
Here, the height is given as 1.5 m and the vertical component of the initial velocity of the piano can be calculated using the formula:
\[ \text{Vertical component of initial velocity} = \text{initial velocity} \times \sin(\text{angle of incline}) \]
Once we have determined the time of flight, that will be the amount of time the workers have to get to the piano before it reaches the bottom of the ramp.