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(1) Consider a medium with a quadratic drag force Fdrag -cv?. || (a) Find an expression for the terminal velocity vr of a projectile in this medium (b) Show that the equation of motion for vertical motion upwards can be written as i = -g[1+ (v/vr)²]. (c) As an intermediate step for part (d), use the chain rule of calculus to show that i = dv/dt can be written as v = v dv/dy. (d) Use the result from part (c) to write the equation of motion of part (b) in terms of dv/dy, and then use separation of variables to find an expression for v(y), i.e. the upwards velcity as a function of y. Assume the initial upwards speed to be ve and the initial position to be y = 0. %3D

(1) Consider a medium with a quadratic drag force Fdrag -cv?. || (a) Find an expression for the terminal velocity vr of a projectile in this medium (b) Show that the equation of motion for vertical motion upwards can be written as i = -g[1+ (v/vr)²]. (c) As an intermediate step for part (d), use the chain rule of calculus to show that i = dv/dt can be written as v = v dv/dy. (d) Use the result from part (c) to write the equation of motion of part (b) in terms of dv/dy, and then use separation of variables to find an expression for v(y), i.e. the upwards velcity as a function of y. Assume the initial upwards speed to be ve and the initial position to be y = 0. %3D

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(1) Consider a medium with a quadratic drag force \( F_{\text{drag}} = -cv^2 \). (a) Find an expression for the terminal velocity \( v_T \) of a projectile in this medium. (b) Show that the equation of motion for vertical motion upwards can be written as \[ \dot{v} = -g \left[ 1 + (v/v_T)^2 \right]. \] (c) As an intermediate step for part (d), use the chain rule of calculus to show that \( \dot{v} \equiv dv/dt \) can be written as \( \dot{v} = v \, dv/dy \). (d) Use the result from part (c) to write the equation of motion of part (b) in terms of \( dv/dy \), and then use separation of variables to find an expression for \( v(y) \), i.e., the upwards velocity as a function of \( y \). Assume the initial upwards speed to be \( v_0 \) and the initial position to be \( y = 0 \).
Transcribed Image Text:(1) Consider a medium with a quadratic drag force \( F_{\text{drag}} = -cv^2 \). (a) Find an expression for the terminal velocity \( v_T \) of a projectile in this medium. (b) Show that the equation of motion for vertical motion upwards can be written as \[ \dot{v} = -g \left[ 1 + (v/v_T)^2 \right]. \] (c) As an intermediate step for part (d), use the chain rule of calculus to show that \( \dot{v} \equiv dv/dt \) can be written as \( \dot{v} = v \, dv/dy \). (d) Use the result from part (c) to write the equation of motion of part (b) in terms of \( dv/dy \), and then use separation of variables to find an expression for \( v(y) \), i.e., the upwards velocity as a function of \( y \). Assume the initial upwards speed to be \( v_0 \) and the initial position to be \( y = 0 \).
Expert Solution
Step 1

(a)

Introduction:

Projectile motion is the motion of an object thrown or projected into the air, subject to only the acceleration of gravity. The object is called a projectile, and its path is called its trajectory.

Calculation:

Write the condition to calculate the terminal velocity. 

W=Fdrag

Here, W is the force of gravity.

Substitute -mg for W where m is the mass of the object and g is the acceleration due to gravity and -cvT2 for Fdrag. Here, the negative sign indicates that the force of gravity is in the downward direction.

-mg=-cvT2vT=mgc

Here, vT is the terminal velocity.

Thus, the expression for the terminal velocity is mgc.

Step 2

(b)

Calculation:

mdvdt=-mg-cv2v˙=-g-cmv2

Substitute gvT2 for cm from the expression of vT in the above expression.

v˙=-g-gvT2v2v˙=-g1+vvT2

Thus, the above expression is the required equation of motion. 

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