Motion in a Plane | Class 11 Physics Notes

3.9 Projectile Motion

Projectile motion is the motion of an object thrown into the air and moving under the influence of gravity alone, neglecting air resistance.

The motion can be analyzed as two independent components: horizontal motion with constant velocity and vertical motion with constant acceleration due to gravity.

If the projectile is launched with initial velocity v₀ at an angle θ₀ with the horizontal (x-axis), the initial velocity components are: v_{0x} = v₀ cos θ₀, v_{0y} = v₀ sin θ₀.

The acceleration vector is a = -g ĵ (downward).

The position coordinates at time t are: x = v_{0x} t = v₀ cos θ₀ t, y = v_{0y} t - (1/2) g t² = v₀ sin θ₀ t - (1/2) g t².

The velocity components at time t are: v_x = v_{0x} = v₀ cos θ₀ (constant), v_y = v_{0y} - g t = v₀ sin θ₀ - g t.

Eliminating time t gives the equation of the trajectory: y = (tan θ₀) x - (g / (2 v_{0x}²)) x², which is a parabola.

Key quantities include:

• Time to reach maximum height: t_m = (v₀ sin θ₀) / g.
• Maximum height: h_m = (v₀² sin² θ₀) / (2 g).
• Time of flight: T_f = 2 t_m = (2 v₀ sin θ₀) / g.
• Horizontal range: R = (v₀² sin 2θ₀) / g.

Examples illustrate calculation of these quantities and verify Galileo's statement about equal ranges for complementary angles around 45°.

📊 Diagram: Fig. 3.16 Motion of an object projected with velocity v₀ at angle θ₀; Fig. 3.17 The path of a projectile is a parabola.

🔗 Connection: This section leads to the study of uniform circular motion, another important type of planar motion.