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).

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�. 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.

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� 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.

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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.

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