To determine the magnitude, angle, and initial velocity of Liza and her dog Jazz's jump across a 150 m wide canyon, we can use the equations of projectile motion. Assuming they follow a parabolic trajectory, we can break down the problem as follows:
Horizontal Motion: The horizontal motion is unaffected by gravity, so we can use the equation:
Horizontal distance = Horizontal velocity × Time
Given that the canyon width is 150 m and the time of flight is 10 seconds, we can calculate the horizontal velocity:
150 m = Horizontal velocity × 10 s
Horizontal velocity = 150 m / 10 s = 15 m/s
Vertical Motion: The vertical motion is affected by gravity. We can use the equation of motion to determine the initial vertical velocity:
Vertical distance = (Initial vertical velocity × Time) + (0.5 × Acceleration due to gravity × Time²)
Since the vertical distance is zero (assuming they land at the same height as takeoff), we can rearrange the equation:
0 = (Initial vertical velocity × 10 s) + (0.5 × 9.8 m/s² × (10 s)²)
Solving for the initial vertical velocity:
Initial vertical velocity = -0.5 × 9.8 m/s² × (10 s) = -49 m/s
Note: The negative sign indicates that the initial vertical velocity is in the opposite direction of the gravitational acceleration (upward).
Magnitude and Angle: To find the magnitude and angle of the initial velocity, we can use the Pythagorean theorem and trigonometric functions:
Magnitude of initial velocity = √[(Horizontal velocity)² + (Initial vertical velocity)²] Angle = arctan(Initial vertical velocity / Horizontal velocity)
Substituting the values:
Magnitude of initial velocity = √[(15 m/s)² + (-49 m/s)²] ≈ 51.23 m/s Angle = arctan((-49 m/s) / (15 m/s)) ≈ -73.93 degrees
The magnitude of the initial velocity is approximately 51.23 m/s, and the angle (measured counterclockwise from the horizontal) is approximately -73.93 degrees. The negative angle indicates that the trajectory is directed downward with respect to the horizontal axis.