Spiderman is standing on the edge of a rooftop 100m above a swimming pool. He decides to jump in to cool off, so
he springs out at a velocity of 30 m/s at an angle of 35 degrees above the horizontal. The near edge pool is
located 127 m from the base of the building. It is a circular pool, with a diameter of 10 m.
a) How high above the rooftop did he jump?
b) What was his acceleration at the apex of his jump?
c) How long was he in the air?
d) What is his vertical velocity just prior to landing?
e) Where does he land relative to the building? Does he end up in the pool?
Did any body solve this yet?
To solve these questions, we can break down the problem into smaller parts and apply the laws of physics. Let's solve each question step by step:
a) To find out how high above the rooftop Spiderman jumped, we need to analyze the vertical component of his initial velocity. We can use the equation:
v^2 = u^2 + 2as
where v is the final velocity, u is the initial velocity, a is the acceleration, and s is the height.
In this case, the final velocity (v) would be 0 m/s because Spiderman comes to a stop at the highest point of his jump. The initial velocity (u) is given as 30 m/s at an angle of 35 degrees above the horizontal. The vertical component of the initial velocity can be found by multiplying the magnitude (30 m/s) by the sine of the angle (35 degrees).
So, the initial vertical velocity (u_y) = 30 m/s * sin(35 degrees).
Now, we can rearrange the equation to solve for the height (s):
s = (v^2 - u^2) / (2a)
Substituting the given values:
s = (0 - (30 m/s * sin(35 degrees))^2) / (2 * -9.8 m/s^2) [Acceleration due to gravity is -9.8 m/s^2.]
Once you calculate the value of 's', you will get the height above the rooftop that Spiderman jumped.
b) To determine the acceleration at the apex of the jump, we need to consider that Spiderman reaches maximum height when his vertical velocity becomes zero. At this point, the only force acting on him is gravity, leading to acceleration due to gravity. So, the acceleration at the apex can be taken as the acceleration due to gravity, which is approximately -9.8 m/s^2.
c) To find out how long Spiderman was in the air, we can use the equation:
s = ut + 0.5at^2
where 's' is the vertical displacement (height), 'u' is the initial vertical velocity, 'a' is the acceleration, and 't' is the time.
Since Spiderman jumps off the rooftop and lands back on the ground (assuming the ground level is the same as the base of the building), his displacement 's' is the height of the rooftop. We already have the values for 'u' (initial vertical velocity) and 'a' (acceleration due to gravity). We need to solve for 't'. Rearranging the equation, we get:
t = sqrt(2s/a)
Substituting the given values, you can calculate the time Spiderman was in the air.
d) To find the vertical velocity just before landing, we can use the equation of motion:
v = u + at
where 'v' is the final velocity, 'u' is the initial velocity, 'a' is the acceleration, and 't' is the time.
In this case, 'u' is the initial vertical velocity (which we found earlier), 'a' is the acceleration due to gravity (-9.8 m/s^2), and 't' is the time (which we found earlier). Substituting these values into the equation, you can calculate the vertical velocity just before landing.
e) To determine where Spiderman lands relative to the building, we can analyze the horizontal motion. Spiderman's initial horizontal velocity remains constant throughout the jump because no horizontal forces (assuming no wind resistance) are acting on him. Therefore, the horizontal distance he travels can be calculated using the equation:
s = ut
where 's' is the horizontal distance, 'u' is the initial horizontal velocity, and 't' is the time (which we found earlier).
The horizontal velocity (u_x) can be found by multiplying the magnitude of the initial velocity (30 m/s) by the cosine of the angle (35 degrees).
So, the initial horizontal velocity (u_x) = 30 m/s * cos(35 degrees).
Once you calculate the horizontal distance traveled (s), you can determine if Spiderman ends up in the pool by comparing it to the distance of the pool from the building's base.
By following these steps and doing the calculations, you should be able to find the answers to all the questions.