lesson .projectilemotion...a projectile withmass m is red initial point (x , y ) angle ofelevation...
TRANSCRIPT
SM��� Calculus III with Optimization Fall ���� Uhan
Lesson ��. ProjectileMotion
� In this lesson...
● Describing the trajectory of a projectile with parametric equations
� Trajectory of a projectile
● A projectile with mass m is �red
○ initial point (x�, y�)○ angle of elevation α○ initial velocity �v�
● Assume:
○ Air resistance is negligible○ �e only external force is due to gravity
x
y �v�
α(x�, y�)
● Let’s derive parametric equations that describe the trajectory of this projectile
�. Let’s de�ne v� = ��v�� (we’re just renaming the initial speed, or themagnitude of the initial velocity). Using thisnew notation, write �v� in terms of v� and α. Hint. We’ll need to use trigonometry.
�. We need an expression for the acceleration �a(t) of the projectile.Recall Newton’s second law ofmotion: if at any time t, a force F(t) acts on an object ofmass m producing anacceleration �a(t), then �F(t) = m�a(t).Since the only external force is due to gravity, which acts downward, we have that �F(t) = m�a(t) = ��,−mg�.Solve for �a(t).
�
�. Using our answer from part �, write an expression for the velocity �v(t) of the projectile.Hint �. Recall that �a(t) = �v′(t). Hint �. Don’t forget the constant vector of integration. Hint �. Since the initialvelocity is �v�, we have �v(�) = �v�. Use the expression for �v� we obtained in part �.
�. Now, using our answer from part �, write an expression for the position �r(t) of the projectile.Hint �. Recall that �v(t) = �r′(t). Hint �. Don’t forget the constant vector of integration. Hint �. Since the initialpoint is (x�, y�), we have �r(�) = �x�, y��.
�. Expand the vector equation we obtained in part � to write parametric equations (i.e. x = . . . , y = . . . ) for thetrajectory of the projectile.
�
� Problems
In each of these problems, ignore the possibility of air resistance. Assume that acceleration due to gravity is downwardand equal to g.
Problem �. A cannon sitting atop of a ���m cli� shoots a projectile at a speed of ��m/s and at an angle of ��° abovethe horizontal. A building ��m tall sits ���m from the base of the cli�. Does the projectile strike the building? (Ignorethe width of the building).
Problem �. A lacrosse player ��m from an open goal throws a ball at an angle of ��° above the horizontal with aspeed of ��m/s. Does the ball enter the goal in the air? Assume that the ball leaves the stick �m above the ground andthat a lacrosse goal is �m high.
�
Problem �. An F-�� is �ying at ���m/s at an altitude of ����m and at an angle of �° below the horizontal when itdrops a bomb. �ere is a ���m building ����m from the point below the F-�� when it drops its bomb. Does the bombhit the building? (Ignore the width of the building.)
Problem �. In Fenway Park, the Green Monster is a wall approximately ��.�m tall and ��m from home plate alongthird base line. A ball was hit at an angle of ��° along the third base line and barely cleared the Green Monster. At whatspeed did the ball leave the bat? Assume that batter hit the ball �m above the ground.
�