A piano has been pushed to the top of the ramp at the back of a moving van. The workers think it is safe, but as they walk away, it begins to roll down the ramp. If the back of the truck is 1.0 m above the ground and the ramp is inclined at 20∘, how much time do the workers have to get to the piano before it reaches the bottom of the ramp?

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Navreaiw · Accepted Answer

Step 1  Given values:  h=1.0m  θ=20∘  g=9.81ms2  Step 2  As the piano start to move down the ramp, the acceleration due gravity is:  a=gsin⁡θ.  If we plug in values in above equation, we will get required value of acceleration:  a=(9.81ms2)sin⁡(20∘)  a=(9.81ms2)(0.34202)  a=3.35ms2  Length of the ramp is given as:  sin⁡θ=h△x  △x=hsin⁡θ  △x=1.0msin⁡(20∘)  △x=1.0m0.34202  △x=2.92m  So, the time before the piano reaches the ground is:  t=△xa  t=2.92m3.35ms2  t=1.32s

Tiefdruckot · Accepted Answer

Step 1  The piano starts to move down the ramp, so the acceleration due to gravity is given by,  a=gsin⁡θ  =(9.80ms2)sin⁡(20∘)  =3.35ms2  Step 2  The length of the ramp is,  Substitute the values,  sin⁡θ=hl  l=hsin⁡θ  l=1.0msin⁡(20∘)  =1.0m0.342  =2.92m  Step 3  According to the equation of motion, the time taken for the piano to reach down the incline is,  s=ut+12at2  s=0+12at2  t=2sa  Step 4  Substitute the values,  t=2(2.92m)(3.35ms2)  =1.32s

user_27qwe · Accepted Answer

First, we need to find the horizontal distance traveled by the piano while rolling down the ramp. This distance can be determined using trigonometry. The horizontal component of the displacement can be calculated as:Δx={ramp&amp;nbsp;length}×cos({ramp&amp;nbsp;angle})Given that the back of the truck is 1.0 m above the ground and the ramp is inclined at 20 degrees, we can calculate the ramp length as follows:{ramp&amp;nbsp;length}=back&amp;nbsp;of&amp;nbsp;truck&amp;nbsp;heightsinramp&amp;nbsp;angleSubstituting the values, we have:{ramp&amp;nbsp;length}=1.0sin(20∘)Now, we can find Δx:Δx=1.0sin(20∘)×cos(20∘)Next, we need to determine the time it takes for the piano to travel this horizontal distance. Since the piano is moving under the influence of gravity, we can use the equation of motion:Δx=12×a×t2where Δx is the displacement, a is the acceleration (due to gravity), and t is the time. The acceleration due to gravity is approximately 9.8 m/s2.We can rearrange the equation to solve for t:t=2×ΔxaSubstituting the values, we have:t=2×(1.0sin(20∘)×cos(20∘))9.8Calculating this expression will give us the time t that the workers have to get to the piano before it reaches the bottom of the ramp.

alenahelenash · Accepted Answer

The key idea is that the piano will slide down the ramp due to its own weight, and we need to find the time it takes for the piano to reach the bottom of the ramp. We can break the weight of the piano into two components: one parallel to the ramp (down the slope) and the other perpendicular to the ramp.The component of the weight parallel to the ramp can be found using the equation:Fparallel=m·g·sin(θ)where m is the mass of the piano and g is the acceleration due to gravity (9.8m/s2).The acceleration of the piano down the ramp can be found using Newton&#039;s second law:Fparallel=m·aparallelSimplifying, we have:aparallel=g·sin(θ)Now, let&#039;s find the time it takes for the piano to travel the distance along the ramp. The distance along the ramp can be calculated using trigonometry:dramp=h·sin(θ)The time it takes for the piano to travel this distance can be found using the equation of motion:d=v0·t+12·a·t2where d is the distance, v0 is the initial velocity (which is zero in this case), a is the acceleration, and t is the time.In this case, d=dramp and a=aparallel. Plugging in these values, we get:dramp=12·g·sin(θ)·t2Now, let&#039;s solve this equation for t. Dividing both sides by 12·g·sin(θ), we have:t2=2·drampg·sin(θ)Finally, taking the square root of both sides, we can find the time t:t=2·drampg·sin(θ)Now we can substitute the given values to find the time. Plugging in h=1.0m and θ=20∘, we have:t=2·1.0·sin(20∘)9.8·sin(20∘)Simplifying further, we get:t=29.8≈0.451sTherefore, the workers have approximately 0.451 seconds to get to the piano before it reaches the bottom of the ramp.

star233 · Accepted Answer

Result:t≈1.32sSolution:To solve this problem, we can use the principles of motion and trigonometry. Let&#039;s denote the time the workers have to get to the piano before it reaches the bottom of the ramp as t.First, we need to find the distance covered by the piano along the ramp. We can use the formula for the displacement along an inclined plane:displacement=length&amp;nbsp;of&amp;nbsp;the&amp;nbsp;ramp×sin(angle&amp;nbsp;of&amp;nbsp;inclination)Given that the length of the ramp is equal to the height of the back of the truck, which is 1.0 m, and the angle of inclination is 20 degrees, we have:displacement=1.0m×sin(20∘)Next, we need to find the time it takes for the piano to cover this displacement along the ramp. We can use the equation of motion:displacement=12×acceleration×time2In this case, the acceleration along the ramp is due to gravity and can be calculated as:acceleration=gravitational&amp;nbsp;acceleration×sin(angle&amp;nbsp;of&amp;nbsp;inclination)The gravitational acceleration is approximately 9.8m/s2. Plugging in the values, we have:1.0m×sin(20∘)=12×(9.8m/s2×sin(20∘))×t2Simplifying the equation, we can cancel out sin(20∘) on both sides:1.0m=12×9.8m/s2×t2Solving for t2, we get:t2=1.0m4.9m/s2Taking the square root of both sides, we find:t=1.0m4.9m/s2Calculating this value, we get:t≈1.32sTherefore, the workers have approximately 1.32 seconds to get to the piano before it reaches the bottom of the ramp.

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