1) The best leaper in the animal kingdom is the puma, which can jump to a height of 3.7 m when leaving the ground at an angle of 45°. With what speed must the animal leave the ground to reach that height? (m/s)
2) A brick is thrown upward from the top of a building at an angle of 15° to the horizontal and with an initial speed of 14 m/s. If the brick is in flight for 3.0 s, how tall is the building? (m)
3) A fireman d = 44.0 m away from a burning building directs a stream of water from a ground-level fire hose at an angle of θi = 26.0° above the horizontal as shown in the figure. If the speed of the stream as it leaves the hose is vi = 40.0 m/s, at what height will the stream of water strike the building? (m)
4) How long does it take an automobile traveling in the left lane of a highway at 50.0 km/h to overtake (become even with) another car that is traveling in the right lane at 35.0 km/h when the cars' front bumpers are initially 115 m apart? (s)
To answer these questions, we can use basic principles of physics and kinematics. Specifically, we can apply the equations of motion to solve for the required quantities.
1) To find the speed at which the puma must leave the ground, we can use the equation for the range of a projectile:
Range = (initial velocity^2 * sin(2θ)) / g
In this case, the range is the vertical height reached by the puma, which is 3.7 m. Since the angle is given as 45°, we can substitute the values into the equation:
3.7 = (v^2 * sin(90)) / g
Simplifying the equation, we get:
v^2 = (3.7 * g) / sin(90)
v = √((3.7 * g) / sin(90))
Substituting the value of the acceleration due to gravity, g = 9.8 m/s^2, we can calculate the speed.
2) To determine the height of the building, we can use the equation for the vertical displacement of a projectile:
Vertical displacement = (initial vertical velocity * time) + (0.5 * acceleration * time^2)
Since the brick is thrown upward, the initial vertical velocity is given by the equation:
Vertical velocity = initial velocity * sin(θ)
Substituting the given values, we can calculate the height of the building.
3) In this scenario, we can break the velocity vector of the water stream into horizontal and vertical components. The vertical component will determine the height at which the water strikes the building. We can use the equations of motion to solve for the vertical displacement:
Vertical displacement = (initial vertical velocity * time) + (0.5 * acceleration * time^2)
The initial vertical velocity can be calculated using the equation:
Vertical velocity = initial velocity * sin(θ)
Substituting the given values, we can calculate the height at which the water strikes the building.
4) To determine the time it takes for the overtaking car to catch up with the other car, we can consider the relative velocity between the two cars. The relative velocity is the difference in their velocities. Using the equation:
Distance = Relative velocity * Time
We can solve for time by rearranging the equation:
Time = Distance / Relative velocity
Substituting the given values, we can calculate the time it takes for the overtaking car to catch up.