Tuesday, August 24, 2010

Velocity Problem Solving

1.      The velocity of a particle is given by v = 20t2 – 100t + 50, where v is in meters per second and t is in seconds. Plot the velocity v and acceleration a versus time for the first 6 seconds of motion and evaluate the velocity when a is zero.

2.      The displacement of a particle is given by s  = 2t2 -30t2 + 100t -50, where s is in meters and t is in seconds. Plot the displacement, velocity, and acceleration as function of time for the first 12 seconds of motion. Determine the time at which the velocity is zero.

3.      The displacement of a particle is given by s = (-2 + 3t) e-0.5t, where s is in meters and t is in seconds. Plot the displacement, velocity, and acceleration versus time for the first 20 seconds of motion. Determine the time at which the acceleration is zero.

Answer:      v = (4 – 1.5 t) e-0.5t

a = (-3.5 + 0.75t) e-0.5t, t = 4.67 s

4.      The velocity of a particle that moves along the s axis is given by v = 2 + 5t3/2, where t is in seconds and v is in meters per second. Evaluate the displacement s, velocity v, and acceleration a when t = 4s. the particle is at the origin s = 0 when t = 0.

5.      the acceleration of a particle is given by a = 4t – 30, where a is in meters per second squared and t is in seconds. Determine the velocity and displacement as functions of time. The initial displacement at t = 0 is s0 = -5 m, and the initial velocity is v0 = 3 m/s.

Answer:      v = 3 – 30t + 2t2

s =  -5 + 3t – 15t2 + 2/3 t3

6.      Calculate the constant acceleration a in g’s which the catapult of an aircraft carrier must provide to produce a launch velocity of 290 km/h in a distance of 100 m. Assume that the carrier is at anchor.

7.      In the final stages of a moon landing, the lunar module descends under retrothrust of its descend engine to which h = 5 m of the lunar surface where it has a downward velocity of 2 m/s. If the descent engine is cut off abruptly at this point, compute the impact velocity of the landing gear with the moon. Lunar gravity is ½ of the earth gravity.

8.      A car comes to a complete stop from an initial speed of 80 km/h in a distance of 30 m. with the same constant acceleration, what would be the stopping distance s from an initial speed of 110 km/h?

9.      A projectile is fired vertically with an initial velocity of 200 m/s. Calculate the maximum altitude h reached by the projectile and the time t after firing for it to return to the ground. Neglect air resistance and take the gravitational acceleration to be constant at 9.81 m/s2.

Answer: h = 2040 m, t = 40.8 s

10.  The graph shows the displacement time history for the rectilinear motion of a particle during an S second interval. Determine the average velocity vav during the interval and to within reasonable limits of accuracy, find the instantaneous velocity v when t = 4 s.

11.  During an 8-second interval the velocity of a particle moving in a straight line varies with the time as shown. Within reasonable limits of accuracy, determine the amount Da by which the acceleration at t = 4 s exceed the average acceleration during the interval. What is the displacement Ds during the interval?

Answer: Da = 0.50 m/s2, Ds = 64 m.

12.  Experimental data for the motion of a particle along a straight line yield measured values of the velocity v for various position coordinates s. A smooth curve is drawn through the points as shown in the graph. Determine the acceleration of the particle when s = 20 m.