Math Problem Statement

The projectile is fired with an initial speed of 25.0 m/s oriented at 37° above the horizontal. (a) Find the position and velocity of the ball when t = 2.00 s. (b) Find the time when the ball reaches the highest point of its flight and height at this point. (c) Find the total time of flight. (d) Find the horizontal range R.

Solution

The image shows a projectile motion problem. Here's a breakdown of the questions in the image:

  1. Projectile initial conditions:
    • Initial speed: v0=25.0m/sv_0 = 25.0 \, \text{m/s}
    • Angle of launch: θ=37\theta = 37^\circ
    • Initial position: (x0,y0)=(0,0)(x_0, y_0) = (0, 0)

The questions are as follows:

(a) Find the position and the velocity of the ball at t=2.00st = 2.00 \, \text{s}.

(b) Find the time when the ball reaches the highest point of its flight and the height at this point.

(c) Find the total time of flight.

(d) Find the horizontal range RR.

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Math Problem Analysis

Mathematical Concepts

Kinematics
Projectile Motion

Formulas

v_x = v_0 * cos(θ)
v_y = v_0 * sin(θ) - g * t
x = v_0 * cos(θ) * t
y = v_0 * sin(θ) * t - 0.5 * g * t^2
Time to highest point: t = (v_0 * sin(θ)) / g
Total time of flight: t_total = 2 * (v_0 * sin(θ)) / g
Range: R = (v_0^2 * sin(2θ)) / g

Theorems

Newton's Second Law of Motion
Equations of Motion

Suitable Grade Level

Grade 11-12, Introductory College Level Physics