The labeled graphs below represent the upward trajectories of 4 bodies, two slid

Question

The labeled graphs below represent the upward trajectories of 4 bodies, two sliding upwards on frictionless inclines and two in free flight. Note that all 4 bodies reach the same maximum height (9 meters) after traveling from the same initial elevation (0 m).

• The initial speed of P is  greater than less than equal to  that of Q.
• The time of travel of P is  greater than less than equal to  that of N.
• The time of travel of R is  greater than less than equal to  that of N
• The initial speed of Q is  greater than less than equal to  that of N.
• The initial speed of N is  greater than less than equal to  that of R.
• The time of travel of P is  greater than less than equal to  that of Q

Explanation / Answer

Before starting, there are some simple observations and deductions to make:

As the graphs are of y vs. x, each simply shows the shape of the path followed by each body.
P is sliding up a steep incline and Q is sliding up a less steep incline.
N and R follow parabolic trajectories - i.e they are simple projectiles.

P and Q must each start with a the same kinetic energy as they have zero kinetic energy at 9m. All of their KE is turned to PE.

N and R are must have equal vertical components of initial velocity to reach the same height. They will reach 9m after the same times, as the vertical motion of projectiles with the same initial vertical components of velocity are identical. However R has a bigger horizontal component of velocity than N - so R travels further in the time needed to reach max height.
There are different ways to explain these - all of them are messy to put into words, but I'll try. I'm assuming you are familiar with kinematics, projectiles and of conservation of mechanical energy.

Q1. equal to

As explained above - they have the same initial kinetic energy.

Q2. Will be same (due to same vertical velocity)

Q3. equal to

Explained above

Q4. less than

Q's initial speed was just sufficient to give it the kinetic energy needed to reach 9m
But N's initial vertical component of velocity was just sufficient to give it the kinetic energy needed to reach 9m AND it had a horizontal velocity component. So N had more total kinetic energy than Q. So Q was slower than N

Q5. less than

They have the same vertical velocity component but N has a smaller horizontal component as explained above. So the initial speed (magnitude of initial velocity) of N is smaller than R's.

Q6. less than

Because they have the same average velocity (as each experiences uniform acceleration) but Q has further to go

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