A skier launches off a ski jump with a horizontal velocity of 25.3 m/s (and no vertical velocity component).
a) What is the magnitude of the horizontal component of her velocity the instant before she lands 1.95 s later?
b) What is the magnitude of the vertical component of her velocity the instant before she lands 1.95 s later?
To solve this problem, we can use the equations of motion for projectiles. The horizontal and vertical motions are independent of each other, so we can analyze them separately.
a) To find the horizontal component of the skier's velocity, we can use the equation:
horizontal distance = horizontal velocity × time
Given:
horizontal velocity (Vx) = 25.3 m/s
time (t) = 1.95 s
Using the equation, we can calculate the horizontal distance traveled:
horizontal distance = 25.3 m/s × 1.95 s = 49.34 m
Therefore, the magnitude of the horizontal component of the skier's velocity before landing is 25.3 m/s.
b) Since the vertical velocity component is not given, we need to calculate it using the equation of motion:
vertical distance = initial vertical velocity × time + (1/2) × acceleration × time^2
We know that the skier jumps off the ski jump with no vertical velocity (Vy = 0 m/s). The only force acting on the skier in the vertical direction is gravity, which causes an acceleration (a) of 9.8 m/s^2 downward.
Plugging in the values:
vertical distance = 0 × 1.95 + (1/2) × (-9.8 m/s^2) × (1.95 s)^2
Simplifying the equation:
vertical distance = 0 m + (-9.8 m/s^2) × (1.95 s)^2
vertical distance = -18.045 m
The negative sign indicates that the skier is below the starting point.
Now, let's find the vertical component of the skier's velocity at the given time using the equation:
vertical velocity = initial vertical velocity + acceleration × time
initial vertical velocity = 0 m/s
acceleration = -9.8 m/s^2 (as it acts in the opposite direction of our positive orientation)
time = 1.95 s
Plugging in the values:
vertical velocity = 0 m/s + (-9.8 m/s^2) × 1.95 s
vertical velocity = -19.11 m/s
Therefore, the magnitude of the vertical component of the skier's velocity just before she lands is approximately 19.11 m/s upward (since we took the downward direction as negative, the magnitude is positive).