A) 13.2 m
The motion of the skier is a projectile motion, which consists of two independent motions:
- a horizontal motion with constant speed
- a vertical motion with constant acceleration [tex]g=-9.8 m/s^2[/tex] (acceleration of gravity) downward
To find the maximum height of her trajectory, we are only concerned with her vertical motion.
The initial vertical velocity upward is
[tex]u_y = u_0 sin \theta = (25) sin 40^{\circ} =16.1 m/s[/tex]
then we can use the following SUVAT equation:
[tex]v_y^2 - u_y^2 = 2ah[/tex]
where
[tex]v_y=0[/tex] is the final vertical velocity, which is zero at the maximum height
[tex]u_y = 16.1 m/s[/tex] is the initial vertical velocity
[tex]a=g=-9.8 m/s^2[/tex]
h is the maximum height
Solving for h,
[tex]h=\frac{v_y^2-u_y^2}{2g}=\frac{-(16.1)^2}{2(-9.8)}=13.2 m[/tex]
B) 1.64 s
The time needed to reach the highest point can be found by analyzing again the vertical motion only. In fact, we can use the following SUVAT equation:
[tex]v_y = u_y +at[/tex]
where
[tex]u_y = 16.1 m/s[/tex]
[tex]a=g=-9.8 m/s^2[/tex]
At the maximum height, the vertical velocity is zero:
[tex]v_y=0[/tex]
So we can solve the equation to find the corresponding time:
[tex]t=\frac{v_y-u_y}{a}=\frac{0-16.1}{-9.8}=1.64 s[/tex]
