ENPH 131 Assignment # Solutions Tutorial Problem (Rocket Height) A rocket, initially at rest on the ground, accelerates straight upward with a constant acceleration of 3. m s. The rocket accelerates for a period of 9. s before exhausting its fuel. The rocket continues its acent until its motion is halted by gravity. The rocket then enters free fall. Find the maximum height, y max, reached by the rocket. Ignore air resistance and assume a constant acceleration due to gravity of 9.81 m s. For consistency in labeling, let us define y s. During the first portion of the trip (first 9s), while the rocket is being propelled, v o, a 39. m s, y o. Apply eqn. 1-5: y y o v o t 1 a t y 1 1 39. 9 y 1 1587.6 m Also, use eqn. 1-4 to calculate the velocity of the rocket at the time the fuel runs out: v v o a t v 1 39. 9 v 1 35.9 m s For the second portion of the trip, from the time the fuels runs out until the rocket reaches its maximum heigh (when v ), y o y 1 1587.6 m, v o v 1 35.8 m s, v, a 9.81 m s. Apply eqn. 1-6: v v o a y y o 35.8 9.81 y max 1587.6 y max 7931.5 m 793 m Problem 1.3 Ball A is thrown vertically upward from the top of a 5 m-high-building with an initial velocity of 7 m s. At the same instant another ball, B, is thrown upward from the ground with an initial velocity of 19 m s. a) Determine the height from the ground at which they pass each other. Take the ground to be s. Ball A: s o 5 m, v o 7 m s, a 9.81 m s. Apply eqn. 1-5: s A 5 7 t 1 9.81 t
ENPH 131 assignment solutions.nb s A 5 7 t 1 9.81 t Ball B: s o, v o 19 m s, a 9.81 m s. Apply eqn. 1-5: s s o v o t 1 a t s B 19 t 1 9.81 t When the balls pass each other: s A s B 5 7 t 1 9.81 t 19 t 1 9.81 t 1 t 5 t.8 s To obtain the height when the balls pass each other, plug t into the eqn. for s B : s s B 19.8 1 9.81.8 s 18.3 m b) Determine the time at which they pass each other. As shown above, the balls pass each other at t.8 s. Problem 1.33 A motorcycle starts from rest at t and travels along a straight road with a constant acceleration of 6 ft s until it reaches a speed of 5 ft s. Afterwards it maintains this speed. Also, when t, a car located 6 ft down the road is traveling toward the motorcycle at a constant speed of 6 ft s. a) Determine the time when they pass each other. For the first portion of the trip, as the motorcycle accelerates: Motorcycle: v o, v 5 ft s, a 6 ft s. Apply eqn. 1-4: v v o a t 5 6 t 1 t 1 8.333 s Thus, the accelerating portion of the trip takes t 1 8.333 s. At this time, t 1, let us calculate the position of the motorcycle and car using eqn. 1-5: Motorcycle: s s o v o t 1 a t s m 1 6. 8.333 s m 8.3 m Car: s c 6 6 8.333 s c 55 m
ENPH 131 assignment solutions.nb 3 s c 6 6 8.333 s c 55 m After t 8.333 s both vehicles are travelling with constant velocities: v m 5 ft s, v c 6 ft s. Use eqn. 1-5 to determine the time, t, when they pass (Note: t is the time after t 1, the initial acceleration period): Motorcycle: Car: s m 8.3 5 t When the vehicles pass each other s m s c : s c 55 6 t Thus, from t, the time when the vehicles pass each other is s m s c 8.3 5 t 55 6 t t 48.11 s t t 1 t t 8.333 48.11 t 56.4 s b) Determine the distance travelled by the motorcycle when they pass each other. Plug t into the eqn. for s m : s m 8.3 5 t s m 8.3 5 48.11 s m 614 m 61 m Tutorial Problem (Graph of Sports Car s Velocity) As shown, the velocity v of a sports car is graphed as a function of time t.
4 ENPH 131 assignment solutions.nb a) Find the sports car s maximum velocity, v max, during the 1s interval depicted in the graph. From the graph above we see that the car s maximum velocity occurs at t 4 s and is roughly 55 m s. v max 55 m s b) During which time interval is the acceleration positive? Acceleration is given by the derivative of velocity with respect to time. On the graph avobe, this is the slope. We see that from t t 4 s the slope of the curve is positive, where as from t 4 s t 1 s the slope is negetive. Therefore the time interval during which acceleration is positive is a for t t 4 s c) Find a max, the magnitude of the sports car s maximum acceleration. The maximum acceleration (in magnitude) will occur at the section of the graph with the steepest slope (whether positive or negetive). We can see this occurs from t t 1 s when the car accelerates from v to v 3 m s: a max v v o 3 t t o 1 a max 3 m s d) Find a min, the minimum magnitude of the sports car s acceleration. The minimum acceleration (in magnitude) will occur at the section of the graph with the shallowest slope. We can see this occurs at t 4 s where, in fact, slope. Therefore a min. m s e) Find the distance r, traveled by the car between t s and t s. From t s t 1 s, v o 3 m s, a 3 m s. Apply eqn. 1-5: s 1 1 3 1 s 1 15 m
ENPH 131 assignment solutions.nb 5 s s o v o t 1 a t s 1 1 3 1 s 1 15 m From t 1 s t s, s o s 1 15 m, v o v 1 3 m s, a m s. Apply eqn. 1-5: s 15 3 1 1 s 55 m Problem 1.6 A motorcyclist starting from rest travels along a straight road and, for 1s, has an acceleration as shown below. a) Find the distance traveled in the 1s. For the first 6s, a 1 6 t. Use eqn. 1- to obtain an expression for the motorcycle s velocity: a t v t 1 v 6 t t v 1 18 t3 Now use eqn. 1-1 to obtain an expression for the motorcycle s displacement: After 6s: v t s t 1 v 18 t3 t s s 1 7 t4 s s 6 1 7 64 s 6 18 m
6 ENPH 131 assignment solutions.nb s 6 1 7 64 s 6 18 m and v 6 1 18 63 v 6 1 m s For the next 4s the motor cycle has a constant acceleration, a 6 m s. Apply eqn. 1-5: s s 6 v 6 t 1 a t s 18 1 4 1 6 4 s 114 m b) Draw the v t that describes the motion. As shown above the velocity of the motorcycle for the first 6s is After that it is given by v 1 18 t3 v v o a t ' v 6 a t t 6 v 1 6 t 6 Using these two equations we can compile a table for the motorcycle s velocity as a function of time: Time s Velocity m s 1.6.1.4 3 1.5 4 3.6 5 6.9 6 1. 7 18. 8 4. 9 3. 1 36.
ENPH 131 assignment solutions.nb 7